text:
Hush, Little Baby
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"Hush Little Baby" redirects here. For the song by British rapper Wretch 32, see Hush Little Baby (Wretch 32 song). For the novel by Caroline B. Cooney, see Hush Little Baby (novel).
This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (April 2010) (Learn how and when to remove this template message)
"Hush, Little Baby" is a traditional lullaby, thought to have been written in the Southern United States. Like most folk songs, the author and date of origin are unknown. The lyrics promise all kinds of rewards to the child if he or she is quiet. The simple structure allows more verses to be added ad lib.
The song has been performed and recorded by many artists including Joan Baez, Regina Spektor, Nina Simone, The Weavers and the Mormon Tabernacle Choir. Additionally, the song has been adapted into pop songs such as Maurice King's "Hambone," Inez and Charlie Foxx's "Mockingbird" and Bo Diddley's eponymous song "Bo Diddley", as well as Eminem's "Mockingbird."
Aretha Franklin, Carly Simon, James Taylor, Etta James, Taj Mahal and Dusty Springfield have each recorded "Mockingbird", which is an R&B variant of the song. "Mockingbird" was featured humorously as a car travel song in the films National Lampoon's Vacation (1983) and Dumb & Dumber (1994).
"Hush Little Baby" has also been used in the play The Miracle Worker, about Helen Keller and Annie Sullivan.
2016 saw this song recorded by violinist Vov Dylan for the album "World Lullabies" which is acting as a fundraiser for Starlight Foundation.[1]
Lyrics[edit]
There are several versions of the song, but the most common lyrics are:
<poem>
Hush, little Baby, don't say a word, Mama's gonna buy you a Mockingbird.
And if that mockingbird don't sing, Mama's gonna buy you a diamond ring.
And if that diamond ring turns brass, Mama's gonna buy you a looking glass.
And if that looking glass is broke, Mama's gonna buy you a billy goat,
And if that billy goat won't pull, Mama's gonna buy you a cart and a bull.
And if that cart and bull turn over, Mama's gonna buy you a dog named Rover.
And if that dog named Rover won't bark, Mama's gonna buy you a horse and a cart.
And if that horse and cart fall down, You'll still be the sweetest little baby in town.
References[edit]
Jump up ^ "World Lullabies". Sanity. Retrieved 2017-01-27.
External links[edit]
Hush Little Baby lyrics
A sing-along version of Hush Little Baby and its illustrated lyrics
[hide]
v
t
e
Nursery rhymes (list)
"A-Tisket, A-Tasket"
"Baa, Baa, Black Sheep"
"Did You Ever See a Lassie?"
"Hey Diddle Diddle"
"Hickory Dickory Dock"
"Hot Cross Buns"
"Humpty Dumpty"
"I'm a Little Teapot"
"It's Raining, It's Pouring"
"Jack and Jill"
"Little Bo Peep"
"Little Boy Blue"
"Little Jack Horner"
"Little Miss Muffet"
"Mary Had a Little Lamb"
"Mary, Mary, Quite Contrary"
"Old King Cole"
"Old MacDonald Had a Farm"
"Old Mother Hubbard"
"Old Woman Who Lived in a Shoe"
"Peter Piper"
"Polly Put the Kettle On"
"Poor Mary"
"Pop Goes the Weasel"
"Pussy Cat Pussy Cat"
"Rain Rain Go Away"
"Roses Are Red"
"Rub-a-dub-dub"
"Simple Simon"
"Sticks and Stones"
"The Grand Old Duke of York"
"The Queen of Hearts"
"Twinkle, Twinkle, Little Star"
"Two Little Dickie Birds"
Clapping
"A Sailor Went to Sea"
"Down Down Baby"
"Mary Mack"
"Miss Susie"
"Miss Lucy"
"Pat-a-cake"
"Pretty Little Dutch Girl"
"Pease Porridge Hot"
Counting
"Duck, duck, goose"
"Eeny, meeny, miny, moe"
"One, Two, Buckle My Shoe"
"One, Two, Three, Four, Five"
"This Old Man"
"Tinker, Tailor"
Finger-
plays
"Five Little Monkeys"
"Itsy Bitsy Spider"
"Little Robin Redbreast"
"Round and round the garden"
"This Little Piggy"
Games
"London Bridge"
"The Muffin Man"
"Mulberry Bush"
"Oranges and Lemons"
"Ring a Ring o' Roses"
"The Farmer in the Dell"
Riddle
"As I was going to St Ives"
Lullabies
"Hush, Little Baby"
"Rock-a-bye Baby"
Rounds
"Ding Dong Bell"
"Frère Jacques"
"Kookaburra"
"Row, Row, Row Your Boat"
"Scotland's Burning"
"Three Blind Mice"
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text:
Sinai Peninsula
From Wikipedia, the free encyclopedia
Jump to: navigation, search
"Sinai" redirects here. For other uses, see Sinai (disambiguation).
Relief map of the Sinai Peninsula
The Sinai Peninsula or simply Sinai (/ˈsaɪnaɪ/;[1][2] Arabic: سيناء Sīnāʼ ; Egyptian Arabic: سينا Sīna, IPA: [ˈsiːnæ]; Hebrew: סיני Sinai; Coptic: ⲥⲓⲛⲁ) is a peninsula in Egypt, the only part of the country located in Asia. It is situated between the Mediterranean Sea to the north and the Red Sea to the south, and is a land bridge between Asia and Africa. Sinai has a land area of about 60,000 km2 (23,000 sq mi) and a population of approximately 1,400,000 people. Administratively, the Sinai Peninsula is divided into two governorates: the South Sinai Governorate and the North Sinai Governorate. Three other governorates span the Suez Canal, crossing into African Egypt: Suez Governorate on the southern end of the Suez Canal, Ismailia Governorate in the center, and Port Said Governorate in the north.
The Sinai Peninsula has been a part of Egypt from the First Dynasty of ancient Egypt (c. 3100 BC). This comes in stark contrast to the region north of it, the Levant (present-day territories of Syria, Lebanon, Jordan, Israel and Palestine), which, due largely to its strategic geopolitical location and cultural convergences, has historically been the centre of conflict between Egypt and various states of Mesopotamia and Asia Minor. In periods of foreign occupation, the Sinai was, like the rest of Egypt, also occupied and controlled by foreign empires, in more recent history the Ottoman Empire (1517–1867) and the United Kingdom (1882–1956). Israel invaded and occupied Sinai during the Suez Crisis (known in Egypt as the Tripartite Aggression due to the simultaneous coordinated attack by the UK, France and Israel) of 1956, and during the Six-Day War of 1967. On 6 October 1973, Egypt launched the Yom Kippur War to retake the peninsula, which was the site of fierce fighting between Egyptian and Israeli forces. By 1982, as a result of the Israel–Egypt Peace Treaty of 1979, Israel had withdrawn from all of the Sinai Peninsula except the contentious territory of Taba, which was returned after a ruling by a commission of arbitration in 1989.
Today, Sinai has become a tourist destination due to its natural setting, rich coral reefs, and biblical history. Mount Sinai is one of the most religiously significant places in the Abrahamic faiths.
Contents
[hide]
1 Name
2 Geography
2.1 Climate
3 History
3.1 Ancient Egypt
3.2 Assyrian Period
3.3 Achaemenid Persian Period
3.4 Hellenistic Period
3.5 Roman and Byzantine Periods
3.6 Early Muslim Period
3.7 Ayyubid Period
3.8 Mamluk and Ottoman Periods
3.9 British control
3.10 Wars with Israel (1948, 56, 67, 67–70, 73)
3.11 1979 Peace Treaty with Israel and aftermath
3.12 Early 21st Century security issues
4 Demographics
5 Economy
6 See also
7 References
8 Further reading
9 External links
Name[edit]
Mount Sinai (Gabal Musa)
The name Sinai may have been derived from the ancient moon-god Sin[3] or from the Hebrew word Seneh (Hebrew: סֶ֫נֶּה Senneh)[4] The peninsula acquired the name due to the assumption that a mountain near Saint Catherine's Monastery is the Biblical Mount Sinai. However this assumption is contested.[5]
In addition to its formal name, Egyptians also refer to it as Arḍ ul-Fairūz (أرض الفيروز "the land of turquoise"). The ancient Egyptians called it Ta Mefkat, or "land of turquoise".[6]
Geography[edit]
Image from Gemini 11 spacecraft, featuring part of Egypt and the Sinai Peninsula in the foreground and the Levant in the background
Sinai is triangular in shape, with northern shore lying on the southern Mediterranean Sea, and southwest and southeast shores on Gulf of Suez and Gulf of Aqaba of the Red Sea. It is linked to the African continent by the Isthmus of Suez, 125 kilometres (78 mi) wide strip of land, containing the Suez Canal. The eastern isthmus, linking it to the Asian mainland, is around 200 kilometres (120 mi) wide. The peninsula's eastern shore separates the Arabian plate from the African plate.[7]
The southernmost tip is the Ras Muhammad National Park.
Most of the Sinai Peninsula is divided among the two governorates of Egypt: South Sinai (Ganub Sina) and North Sinai (Shamal Sina).[8] Together, they comprise around 60,000 square kilometres (23,000 sq mi) and have a population (January 2013) of 597,000. Three more governates span the Suez Canal, crossing into African Egypt: Suez (el-Sewais) is on the southern end of the Suez Canal, Ismailia (el-Isma'ileyyah) in the centre, and Port Said in the north.
The largest city of Sinai is Arish, capital of the North Sinai, with around 160,000 residents. Other larger settlements include Sharm el-Sheikh and El-Tor, on the southern coast. Inland Sinai is arid, mountainous and sparsely populated, the largest settlements being Saint Catherine and Nekhel.[8]
Climate[edit]
Sinai is one of the coldest provinces in Egypt because of its high altitudes and mountainous topographies. Winter temperatures in some of Sinai's cities and towns reach −16 °C (3 °F).[citation needed]
History[edit]
This section needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (March 2015) (Learn how and when to remove this template message)
Sinai Peninsula in hieroglyphs

Biau
Bj3w[9]
"Mining country"[9]
Ancient Egypt[edit]
Sinai was called Mafkat or "country of turquoise" by the ancient Egyptians [10][11] From the time of the First Dynasty or before, the Egyptians mined turquoise in Sinai at two locations, now called by their Egyptian Arabic names Wadi Magharah and Serabit El Khadim. The mines were worked intermittently and on a seasonal basis for thousands of years. Modern attempts to exploit the deposits have been unprofitable. These may be the first historically attested mines.[citation needed]
Assyrian Period[edit]
Achaemenid Persian Period[edit]
At the end of the time of Darius I, the Great (521–486 BCE) Sinai was part of the Persian province of Abar-Nahra, which means "beyond the river [Euphrates].[12]
Cambyses successfully managed the crossing of the hostile Sinai Desert, traditionally Egypt's first and strongest line of defence, and brought the Egyptians under Psamtik III, son and successor of Ahmose, to battle at Pelusium. The Egyptians lost and retired to Memphis; the city fell to the Persian control and the Pharaoh was carried off in captivity to Susa in mainland Persia.
Hellenistic Period[edit]
Roman and Byzantine Periods[edit]
St. Catherine's Monastery is the oldest working Christian monastery in the world and the most popular tourist attraction on the peninsula.
After the death of the last Nabatean king, Rabbel II Soter, in 106,[13] the Roman emperor Trajan faced practically no resistance and conquered the kingdom on 22 March 106. With this conquest, the Roman Empire went on to control all shores of the Mediterranean Sea. The Sinai Peninsula became part of the Roman province of Arabia Petraea.[14]
Saint Catherine's Monastery on the foot of Mount Sinai was constructed by order of the Emperor Justinian between 527 and 565. Most of the Sinai Peninsula became part of the province of Palaestina Salutaris in the 6th century.
Early Muslim Period[edit]
Ayyubid Period[edit]
During the Crusades it was under control of Fatimid Caliphate. Later, Saladin abolished the Fatimid Caliphate in Egypt and took this region under his control too. It was the military route from Cairo to Damascus during the Crusades.[citation needed]
Mamluk and Ottoman Periods[edit]
The peninsula was governed as part of Egypt under the Mamluk Sultanate of Egypt from 1260 until 1517, when the Ottoman Sultan, Selim the Grim, defeated the Egyptians at the Battles of Marj Dabiq and al-Raydaniyya, and incorporated Egypt into the Ottoman Empire. From then until 1906, Sinai was administered by the Ottoman provincial government of the Pashalik of Egypt, even following the establishment of the Muhammad Ali Dynasty's rule over the rest of Egypt in 1805.
The wilderness of Sinai, 1862
British control[edit]
In 1906, the Ottoman Porte formally transferred administration of Sinai to the Egyptian government, which essentially meant that it fell under the control of the United Kingdom, who had occupied and largely controlled Egypt since 1882. The border imposed by the British runs in an almost straight line from Rafah on the Mediterranean shore to Taba on the Gulf of Aqaba. This line has served as the eastern border of Egypt ever since.
Wars with Israel (1948, 56, 67, 67–70, 73)[edit]
Canadian and Panamanian UNEF UN peacekeepers in Sinai, 1974
At the beginning of the 1948 Arab–Israeli War, Egyptian forces entered the former British Mandate of Palestine from Sinai to support Palestinian and other Arab forces against the newly declared State of Israel. For a period during the war, Israeli forces entered the north-eastern corner of Sinai.[15] With the exception of Palestine's Gaza Strip, which came under the administration of the All-Palestine Government,[16] the western frontier of the former Mandate of Palestine became the Egyptian–Israeli frontier under the 1949 Armistice Agreement. In 1958, the Gaza Strip came under direct Egyptian military administration, though it was governed separately from Sinai, and was never annexed by Egypt. The Egyptian government maintained that Egyptian administration would be terminated upon the end of the conflict with Israel.
In 1956, Egypt nationalised the Suez Canal,[17] a waterway marking the boundary between Egyptian territory in Africa and the Sinai Peninsula. Thereafter, Israeli ships were prohibited from using the Canal,[18] owing to the state of war between the two states. Egypt also prohibited ships from using Egyptian territorial waters on the eastern side of the peninsula to travel to and from Israel, effectively imposing a blockade on the Israeli port of Eilat. Subsequently, in what is known in Egypt as the Tripartite Aggression, Israeli forces, aided by Britain, and France (which sought to reverse the nationalisation and regain control over the Suez Canal), invaded Sinai and occupied much of the peninsula within a few days. Several months later Israel withdrew its forces from Sinai, following strong pressure from the United States and the Soviet Union. Thereafter, the United Nations Emergency Force (UNEF) was stationed in Sinai to prevent any further conflict in the Sinai.
In 1967, Egypt reinforced its military presence in Sinai and on 16 May ordered the UNEF out of Sinai with immediate effect.[19] Secretary-General U Thant eventually complied and ordered the withdrawal without Security Council authorisation. In the course of the Six-Day War that broke out shortly thereafter, Israel captured the entire Sinai Peninsula, and Gaza Strip from Egypt, the West Bank (including East Jerusalem) from Jordan (which it had ruled since 1949), and the Golan Heights from Syria. The Suez Canal, the east bank of which was now occupied by Israel, was closed. Israel commenced efforts at large scale Israeli settlement in the peninsula.
Following the Israeli conquest of Sinai, Egypt launched the War of Attrition (1967–70) aimed at forcing Israel to withdraw from Egyptian territory. The war saw protracted conflict in the Suez Canal Zone, ranging from limited to large scale combat. Israeli shelling of the cities of Port Said, Ismailia, and Suez on the west bank of the canal, led to high civilian casualties (including the virtual destruction of Suez), and contributed to the flight of 700,000[20] Egyptian internal refugees. Ultimately, the war concluded in 1970 with no change in the front line.[21]
On 6 October 1973, Egypt commenced Operation Badr to retake the Sinai, while Syria launched a simultaneous operation to retake the Golan Heights,[citation needed] thereby beginning the Yom Kippur War (known in Egypt as the October War). Egyptian engineering forces built pontoon bridges to cross the Suez Canal, and stormed the Bar-Lev Line, Israel's defensive line along the canal. Though the Egyptians maintained control of most of the east bank of the Canal, in the later stages of the war, the Israeli military crossed the southern section of Canal, cutting off the Egyptian 3rd Army, and occupied a section of the west bank. The war ended following a mutually agreed-upon ceasefire. After the war, as part of the subsequent Sinai Disengagement Agreements, Israel withdrew from the Canal, with Egypt agreeing to permit passage of Israeli ships. The canal was reopened in 1975, with President Sadat leading the first convoy through the canal aboard an Egyptian destroyer.
1979 Peace Treaty with Israel and aftermath[edit]
Egypt-Israel border. Looking north from the Eilat Mountains
In 1979, Egypt and Israel signed a peace treaty in which Israel agreed to withdraw from the entirety of Sinai. Israel subsequently withdrew in several stages, ending in 1982. The Israeli pull-out involved dismantling almost all Israeli settlements, including the settlement of Yamit in north-eastern Sinai. The exception was the coastal city of Sharm el-Sheikh, which the Israelis had founded as Ofira during the period of their occupation. The Treaty allows monitoring of Sinai by the Multinational Force and Observers, and limits the number of Egyptian military forces in the peninsula.
Early 21st Century security issues[edit]
Since the early 2000s, Sinai has been the site of several terror attacks against tourists, the majority of whom are Egyptian. Investigations have shown that these were mainly motivated by a resentment of the poverty faced by many Bedouin in the area. Attacking the tourist industry was viewed as a method of damaging the industry so that the government would pay more attention to their situation.[22] (See 2004 Sinai bombings, 2005 Sharm El Sheikh bombings and 2006 Dahab bombings). Since the 2011 Egyptian Revolution, unrest has become more prevalent in the area including the 2012 Egyptian-Israeli border attack in which 16 Egyptian soldiers were killed by militants. (See Sinai insurgency).
Also on the rise are kidnappings of refugees. According to Meron Estifanos, Eritrean refugees are often kidnapped by Bedouin in the northern Sinai, tortured, raped, and only released after receiving a large ransom.[23][24]
Under President el-Sisi, Egypt has implemented a rigorous policy of controlling the border to the Gaza Strip, including the dismantling of tunnels between Gaza and Sinai.[25]
Demographics[edit]
Two young Bedouins making bread in the desert
The two governorates of North and South Sinai have a total population of 597,000 (January 2013). This figure rises to 1,400,000 by including Western Sinai, the parts of the Port Said, Ismailia and Suez Governorates lying east of the Suez Canal. Port Said alone has a population of roughly 500,000 people (January 2013). Portions of the populations of Ismailia and Suez live in west Sinai, while the rest live on the western side of the Suez Canal.
Population of Sinai has largely consisted of desert-dwelling Bedouins with their colourful traditional costumes and significant culture.[26] Large numbers of Egyptians from the Nile Valley and Delta moved to the area to work in tourism, but development adversely affected the native Bedouin population.[citation needed] In order to help alleviate their problems, various NGOs began to operate in the region, including the Makhad Trust, a UK charity that assists the Bedouin in developing a sustainable income while protecting Sinai's natural environment, heritage and culture.[citation needed]
Economy[edit]
Dahab in Southern Sinai is a popular beach and diving resort
See also: Tourism in Egypt
Since the Israeli–Egyptian peace treaty, Sinai's scenic spots (including coral reefs offshore) and religious structures have become important to the tourism industry. The most popular tourist destination in Sinai are Mount Sinai (Jabal Musa) and St Catherine's Monastery, which is considered to be the oldest working Christian monastery in the world, and the beach resorts of Sharm el-Sheikh, Dahab, Nuweiba and Taba. Most tourists arrive at Sharm el-Sheikh International Airport, through Eilat, Israel and the Taba Border Crossing, by road from Cairo or by ferry from Aqaba in Jordan.[citation needed]
See also[edit]
Isthmus of Suez
Operation Eagle
Multinational Force and Observers
Negev Bedouin
Natural places
Desert of Paran
Mitla Pass
Manmade structures
Biblical Mount Sinai
Nawamis
Wildlife
Sinai leopard
References[edit]
Jump up ^ "Definition of Sinai". The Free Dictionary. Retrieved 2014-05-03.
Jump up ^ "Define Sinayi". Dictionary.com. Retrieved 2014-05-03.
Jump up ^ "Sinai Peninsula (peninsula, Egypt) - Britannica Online Encyclopedia". Britannica.com. Retrieved 14 January 2012.
Jump up ^ "Sinai, Mount". JewishEncyclopedia.com. Retrieved 14 January 2012.
Jump up ^ See Biblical Mount Sinai for a fuller discussion.
Jump up ^ "Étude de la turquoise : de ses traitements et imitations" Archived 15 December 2013 at the Wayback Machine., thesis by Claire Salanne, Université de Nantes, 2009.
Jump up ^ Homberg, Catherine and Martina Bachmann, Evolution of the Levant Margin and Western Arabia Platform Since the Mesozoic, The Geological Society of London, 2010, p 65 ISBN 978-1862393066
^ Jump up to: a b Ned Greenwood (1 January 2010). The Sinai: A Physical Geography. University of Texas Press. pp. 4–. ISBN 978-0-292-77909-9.
^ Jump up to: a b The translation "mining country" is not certain, see also Rainer Hannig: Großes Handwörterbuch Ägyptisch-Deutsch : (2800 - 950 v. Chr.). p. 1135.
Jump up ^ http://www.egyptsearch.com/forums/ultimatebb.cgi?ubb=print_topic;f=8;t=008397
Jump up ^ Joseph Davidovits and Ralph Davidovits (2007). "Why Djoser's blue Egyptian faience tiles are not blue? Manufacturing Djoser's faience tiles at temperatures as low as 250 °C?". In Jean Claude Goyon, Christine Cardin. Proceedings of the ninth International Congress of Egyptologists (PDF). 1. Louvain/Paris/Dudley. p. 375.
Jump up ^ Iranchamber
Jump up ^ Schürer, Emil; Millar, Fergus; Vermes, Geza (26 March 2015). The History of the Jewish People in the Age of Jesus Christ:. Bloomsbury Academic. p. 583. ISBN 978-0-567-50161-5.
Jump up ^ Taylor, Jane: Petra And the Lost Kingdom of the Nabataeans. I. B. Tauris 2001, ISBN 1860645089, p. 73-74 (online copy, p. 73, at Google Books)
Jump up ^ Rogan, Eugene L. and Avi Shlaim, eds. The War for Palestine: Rewriting the History of 1948. 2nd edition. Cambridge: Cambridge UP, p. 99, 2007
Jump up ^ Shlaim, Avi (2001). Israel and the Arab Coalition. In Eugene Rogan and Avi Shlaim (eds.). The War for Palestine (pp. 97). Cambridge: Cambridge University Press. ISBN 978-0-521-79476-3
Jump up ^ http://www.history.com/this-day-in-history/egypt-nationalizes-the-suez-canal
Jump up ^ "1956: Egypt Seizes Suez Canal". BBC. 26 July 1956.
Jump up ^ Samir A. Mutawi (18 July 2002). Jordan in the 1967 War. Cambridge University Press. p. 93. ISBN 978-0-521-52858-0. Although Eshkol denounced the Egyptians, his response to this development was a model of moderation. His speech on 21 May demanded that Nasser withdraw his forces from Sinai but made no mention of the removal of UNEF from the Straits nor of what Israel would do if they were closed to Israeli shipping. The next day Nasser announced to an astonished world that henceforth the Straits were, indeed, closed to all Israeli ships
Jump up ^ Spencer, Tucker. Encyclopedia or the Arab-Israeli Conflict. p. 175.
Jump up ^ "War of Attrition".
Jump up ^ Serene Assir (23 July 2005). "Shock in Sharm". Al-Ahram Weekly. Retrieved 29 March 2014.
Jump up ^ Meron Stefanos on the torture houses in north Sinai
Jump up ^ Sound of Torture documentary
Jump up ^ http://www.al-monitor.com/pulse/originals/2015/04/egypt-sinai-gaza-tunnels-sanctions-sisi-terrorist.html#
Jump up ^ Leonard, William R. and Michael H. Crawford, The Human Biology of Pastoral Populations, Cambridge University Press, 2002, p. 67 ISBN 978-0521780162
Further reading[edit]
Gardner, Ann. "At Home in South Sinai". Nomadic Peoples 2000. Vol. 4, Iss. 2; pp. 48–67. Detailed account of Bedouin women
H. J. L. Beadnell (May 1926). "Central Sinai". Geographical Journal. 67 (5): 385–398. doi:10.2307/1782203. JSTOR 1782203.
C. W. Wilson (1873). "Recent Surveys in Sinai and Palestine". Journal of the Royal Geographical Society of London. 43: 206–240. doi:10.2307/1798627. JSTOR 1798627.
Jacobs, Jessica (2006). "Tourist Places and Negotiating Modernity: European Women and Romance Tourism in the Sinai". In Minca, Claudio; Oakes, Tim. Travels in Paradox: Remapping Tourism. Rowman & Littlefield. ISBN 978-0-7425-2876-5. Retrieved 7 January 2010.
Teague, Matthew; Moyer, Matt (March 2009). "The Sinai's Separate Peace". National Geographic Magazine. Washington, D.C.: National Geographic Society. 215 (3): 99–121. ISSN 0027-9358. Retrieved 7 January 2010.
Jarvis, C.S.,Yesterday and To-day in Sinai (Edinburgh/London: W. Blackwood & Sons, 1931).
New terrorist challenges in the Sinai peninsula, prominent jihadists organisations, Strategic Impact (52), issue: 3 / 2014, pp. 39-47
External links[edit]
Wikimedia Commons has media related to Sinai Peninsula.
Wikivoyage has a travel guide for Sinai Peninsula.
Guide to Sinai, covering background information on history, flora, fauna, desert, Bedouin, safaris and geology of Sinai
Sinai Local Magazine
The Complete Guide To: The Sinai, The Independent, 15 March 2008.
Sinai in ancient Egypt
Broadcasting videos from Sinai
Images of the Sinai Desert
IRIN humanitarian news: EU grant to tackle rural poverty in South Sinai
Sinai trekking and safari: route maps and photo archive
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[show]
v
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People and things in the Quran
[hide]
Characters
Non-humans
Allâh ("The God")
Names of Allah found in the Quran
Beings in Jannah (Paradise, literally 'Garden')
Ghilman & Wildan
Houri
Zaqqum
Animals
Related
The baqarah (cow) of Israelites
The dhi’b (wolf) that Jacob feared could attack Joseph
The hud-hud (hoopoe) of Solomon
The kalb (dog) of the sleepers of the cave
The nūn (fish or whale) of Jonah
The nāqaṫ (she-camel) of Saleh
The fīl (elephant) of the Abyssinians)
Non-related
Ḥimār (domesticated donkey or wild ass)
Qaswarah ('Lion', 'Beast of prey' or 'Hunter')
Jinns
Iblīs the Devil
Ifrit
Marid
Qareen
Prophets
Mentioned
Ādam (Adam)
Al-Yasa‘ (Elisha)
Ayyūb (Job)
Dāwūd (David)
Dhūl-Kifl (Ezekiel?)
Hārūn (Aaron)
Hūd (Eber?)
Idrīs (Enoch?)
Ilyās (Elijah)
‘Imrān (Joachim the father of Maryam)
Is-ḥāq (Isaac)
Ismā‘īl (Ishmael)
Dhabih Ullah
Isma'il Ṣādiq al-Wa‘d (Fulfiller of the Promise)
Lūṭ (Lot)
Ṣāliḥ
Shu‘ayb (Jethro, Reuel or Hobab?)
Sulaymān ibn Dāwūd (Solomon son of David)
‘Uzair (Ezra?)
Yaḥyā ibn Zakariyyā (John the Baptist the son of Zechariah)
Ya‘qūb (Jacob)
Isrâ’îl (Israel)
Yūnus (Jonah)
Dhūn-Nūn ("He of the Fish (or Whale)" or "Owner of the Fish (or Whale)")
Ṣāḥib al-Ḥūṫ ("Companion of the Whale")
Yūsuf ibn Ya‘qūb (Joseph son of Jacob)
Zakariyyā (Zechariah)
Ulu-l-‘Azm
Muḥammad
Aḥmad
Other names and titles of Muhammad
ʿĪsā (Jesus)
Al-Masīḥ (The Messiah)
Ibn Maryam (Son of Mary)
Mūsā Kalīmullāh (Moses He who spoke to God)
Ibrāhīm Khalīlullāh (Abraham Friend of God)
Nūḥ (Noah)
Debatable ones
Dhūl-Qarnain (Cyrus the Great?)
Luqmân
Maryam (Mary)
Ṭâlûṫ (Saul or Gideon?)
Implied
Ermia/Jeremiah
Ṣamû’îl (Samuel)
Yusha' ibn Nūn/Joshua (companion and successor of Musa)
People of Prophets
Mentioned evil ones
Āzar (possibly Terah)
Fir'aun (Pharaoh of Moses)
Hāmān
Jâlûṫ (Goliath)
Qarun/Korah (cousin of Musa/Moses)
As-Sāmirī
Abu Lahab
Specified good ones
Zechariah's wife
Adam's immediate relatives
Martyred son
Wife
Women of Moses
Moses' mother
Sister
Wife
Sister-in-law
Isaac's mother
Zayd
Non-specified good ones
Abiona/Amtelai daughter of Karnebo (mother of Ibrahim)
Bathsheba (wife of Dawud, mother of Sulayman)
Hājar/Hagar (wife of Ibrahim, mother of Isma'il)
Hannah/Anne daughter of Faquz (mother of Maryam, wife of Imran)
Imran/Amram (father of Musa)
Lamech (father of Nuh)
Rāhil/Rachel (wife of Ya'qub, mother of Yūsuf)
Rahma/Dinah (wife of Ayyub)
Shamkha bint Anush/Betenos (mother of Nuh)
Son of Luqman
Implied
Asiyah bint Muzahim/Bithiah (wife of Fir'aun)
Asif ibn Barkhiya (vizier and successor of Sulayman)
Bilquis (Queen of Saba/Sheba)
Believer of Fir'aun Family (Hizbil/Hizqil ibn Sabura)
Benyamin/Benjamin
Habib the Carpenter (believer of Ya-Sin)
Kaleb/Caleb (companion of Yusha')
Khidr
Magicians of Fir'aun
Simon Cephas/Simon Peter
Abrahah
Bal'am/Balaam
Barsisa
Nebuchadnezzar II
Nimrod
Potiphar (Al-Aziz)
Shaddad
Simeon (son of Ya'qub)
Slayers of Saleh's she-camel (Qaddar ibn Salif and Musda' ibn Dahr)
Valid ibn Rayyan (king of Egypt in the account of Yūsuf)
Zuleikha (wife of al-Aziz)
Groups
Mentioned
Aş-ḥāb al-Jannah
People of Paradise
People of the Burnt Garden
Aş-ḥāb as-Sabṫ (Companions of the Sabbath)
Christian apostles
Disciples of Jesus
Companions of Noah's Ark
Aş-ḥāb al-Kahf war-Raqīm (Companions of the Cave and Al-Raqaim?
Companions of the Elephant
People of al-Ukhdūd
People of the City (People of Ya-Sin)
People of Yathrib or Medina
Folk of Dawud
Folk of Lut
Folk of Musa and Harun
Nation of Nuh
Tribes, ethnicities or families
A‘rāb (Arabs or Bedouins)
'Ād (people of Hud)
Companions of the Rass
Qawm Tubba‘ (People of Tubba)
People of Saba'/Sheba
Quraysh
Thamûd (people of Saleh)
Aṣ-ḥâb al-Ḥijr ("Companions of the Stoneland")
Ajam
Ar-Rûm (literally "The Romans")
Banî Isrâ’îl (Children of Israel)
Mu’ṫafikāṫ (The overthrown cities of Sodom and Gomorrah)
People of Ibrahim
People of Ilyas
People of Nuh
People of Shuaib
Ahl Madyan People of Madyan)
Aṣ-ḥāb al-Aykah ("Companions of the Wood")
Qawm Yûnus (People of Jonah)
Ya'juj and Ma'juj/Gog and Magog
Ahl al-Bayṫ ("People of the Household")
Household of Abraham
Brothers of Yūsuf
Daughters of Abraham's nephew Lot (Ritha, Za'ura, et al.)
Progeny of Imran
Household of Moses
Household of Muhammad ibn Abdullah ibn Abdul-Muttalib ibn Hashim
Daughters of Muhammad
Wives of Muhammad
Household of Salih
People of Fir'aun
Current Ummah of Islam (Ummah of Muhammad)
Aṣ-ḥāb Muḥammad (Companions of Muhammad)
Muhajirun (Emigrants)
Anṣār Muslims of Medina who helped Muhammad and his Meccan followers, literally 'Helpers')
People of Mecca
Umm Jamil (wife of Abu Lahab)
Children of Ayyub
Dead son of Sulaiman
Qabil/Cain (son of Adam)
Wali'ah or Wa'ilah/Waala (wife of Nuh)
Walihah or Wahilah (wife of Lut)
Yam or Kan'an (son of Nuh)
Implicitly
mentioned
Amalek
Ahl al-Suffa (People of the Verandah)
Banu Nadir
Banu Qaynuqa
Banu Qurayza
Iranian people
Umayyad Dynasty
Aus & Khazraj
People of Quba
Religious groups
Ahl al-dhimmah (Dhimmi)
Kâfirûn (Infidels)
Zoroastrians
Munafiqun (Hypocrites)
Muslims
People of the Book (Ahl al-Kiṫāb)
Naṣārā (Christian(s) or People of the Injil)
Ruhban (Christian monks)
Qissis (Christian priest)
Yahūd (Jews)
Ahbār (Jewish scholars)
Rabbani/Rabbi
Sabians
Polytheists
Meccan polytheists at the time of Muhammad
Mesopotamian polytheists at the time of Abraham and Lot
Locations
Mentioned
Al-Arḍ Al-Mubārakah ("The Land The Blessed")
Al-Arḍ Al-Muqaddasah ("The Land The Holy")
In the Arabian Peninsula (excluding Madyan)
Al-Aḥqāf ("The Sandy Plains," or "the Wind-curved Sand-hills")
Iram dhāṫ al-‘Imād (Iram of the Pillars)
Al-Madīnah (formerly Yathrib)
‘Arafāṫ
Al-Ḥijr (Hegra)
Badr
Ḥunayn
Makkah (Mecca)
Bakkah
Ka‘bah (Kaaba)
Maqām Ibrāhīm (Station of Abraham)
Safa and Marwah
Saba’ (Sheba)
‘Arim Saba’ (Dam of Sheba)
Rass
In Mesopotamia:
Al-Jūdiyy
Munzalanm-Mubārakan ("Place-of-Landing Blessed")
Bābil (Babylon)
Qaryaṫ Yūnus ("Township of Jonah," that is Nineveh)
Door of Hittah
Madyan (Midian)
Majma' al-Bahrain
Miṣr (Mainland Egypt)
Sinai Region or Tīh Desert
Al-Wād Al-Muqaddas Ṭuwan (The Holy Valley of Tuwa)
Al-Wādil-Ayman (The valley on the 'righthand' side of the Valley of Tuwa and Mount Sinai)
Mount Sinai or Mount Tabor
Mu'tafikat (Sodom)
Implied
Antioch
Antakya
Arabia
Ayla
Barrier of Dhul-Qarnayn
Bayt al-Muqaddas & 'Ariha
Bilād ar-Rāfidayn (Mesopotamia)
Canaan
Cave of Seven Sleepers
Dār al-Nadwa
Al-Ḥijāz (literally "The Barrier")
Black Stone (Al-Ḥajar al-Aswad) & Al-Hijr of Isma'il
Cave of Hira & Ghar al-Thawr (Cave of the Bull)
Ta'if
Hudaybiyyah
Jordan River
Nile River
Palestine River
Paradise of Shaddad
Religious locations
Bay'a (Church)
Mihrab
Monastery
Masjid (Mosque, literally "Place of Prostration")
Al-Mash‘ar Al-Ḥarām ("The Monument the Sacred")
Al-Masjid Al-Aqṣā (Al-Aqsa Mosque, literally "The Place-of-Prostration The Farthest")
Al-Masjid Al-Ḥarām (The Sacred Mosque of Mecca)
Masjid al-Dirar
A Mosque in the area of Medina, possibly:
Masjid Qubâ’ (Quba Mosque)
The Prophet's Mosque
Salat (Synagogue)
Non-human
physical
entities
Religious texts
Al-Injîl (The Gospel of Jesus)
Al-Qur’ân (The book of Muhammad)
Ṣuḥuf-i Ibrâhîm (Scroll(s) of Abraham)
Aṫ-Ṫawrâṫ (The Torah)
Ṣuḥuf-i-Mûsâ (Scroll(s) of Moses)
Tablets of Stone
Az-Zabûr (The Psalms of David)
Related objects
Forbidden fruit of Adam
Heavenly Food of Christian Apostles
Noah's Ark
Staff of Musa
Tabut al-Sakina (Casket of Shekhinah)
Throne of Bilqis
Trumpet of Israfil
Mentioned idols
(cult images)
'Ansāb
Idols of Israelites:
Baal
The ‘ijl (golden calf statue) of Israelites
Idols of Noah's people:
Nasr
Suwā‘
Wadd
Yaghūth
Ya‘ūq
Idols of Quraysh:
Lāt
Manāṫ
'Uzza
Jibt and Taghut
Events
Battle of al-Aḥzāb ("the Confederates")
Battle of Badr
Battle of Hunayn
Battle of Khaybar
Battle of Tabouk
Battle of Uhud
Conquest of Mecca
Incident of Ifk
Laylat al-Mabit
Mubahala
Sayl al-‘Arim (Flood of the Great Dam of Marib in Sheba)
The Farewell Pilgrimage (Hujja al-Wada')
Treaty of Hudaybiyyah
Umrah al-Qaza
Yawm al-Dār
Implied
The event of Ghadir Khumm
Note: The names are sorted alphabetically. Standard form: Islamic name / Biblical name (title or relationship)
Authority control
WorldCat Identities
VIAF: 162783451
GND: 4104369-8
NDL: 00628596
Coordinates: 29°30′N 33°50′E / 29.500°N 33.833°E / 29.500; 33.833
Retrieved from "https://en.wikipedia.org/w/index.php?title=Sinai_Peninsula&oldid=816580927"
Categories:
Sinai Peninsula
Peninsulas of Asia
Peninsulas of Egypt
Geography of Western Asia
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Landforms of Western Asia
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Sinai (disambiguation)

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Abrahamic faiths

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Nabatean Kingdom

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1949 Armistice Agreement

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Bar-Lev Line

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Israeli military

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2011 Egyptian Revolution

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Daughters of Lot

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Ancient Mesopotamian polytheism

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Safa and Marwah

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Al-Munzal Al-Mubârak

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Sinai Peninsula in Islam

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Sayl al-‘Arim

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Inner product space
From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
For the scalar product or dot product of coordinate vectors, see dot product.
See also: Parallelogram law and Polarization identity
Geometric interpretation of the angle between two vectors defined using an inner product
Scalar product spaces, over any field, have "scalar products" that are symmetrical and linear in the first argument. Hermitian product spaces are restricted to the field of complex numbers and have "Hermitian products" that are conjugate-symmetrical and linear in the first argument. Inner product spaces may be defined over any field, having "inner products" that are linear in the first argument, conjugate-symmetrical, and positive-definite. Unlike inner products, scalar products and Hermitian products need not be positive-definite.
In linear algebra, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors. Inner products allow the rigorous introduction of intuitive geometrical notions such as the length of a vector or the angle between two vectors. They also provide the means of defining orthogonality between vectors (zero inner product). Inner product spaces generalize Euclidean spaces (in which the inner product is the dot product, also known as the scalar product) to vector spaces of any (possibly infinite) dimension, and are studied in functional analysis. The first usage of the concept of a vector space with an inner product is due to Peano, in 1898.[1]
An inner product naturally induces an associated norm, thus an inner product space is also a normed vector space. A complete space with an inner product is called a Hilbert space. An (incomplete) space with an inner product is called a pre-Hilbert space, since its completion with respect to the norm induced by the inner product is a Hilbert space. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces.
Contents
1 Definition
1.1 Alternative definitions, notations and remarks
1.2 Elementary properties
2 Examples
2.1 Real numbers
2.2 Euclidean space
2.3 Complex coordinate space
2.4 Hilbert space
2.5 Random variables
2.6 Real matrices
2.7 Vector spaces with forms
3 Norms on inner product spaces
4 Orthonormal sequences
5 Operators on inner product spaces
6 Generalizations
6.1 Degenerate inner products
6.2 Nondegenerate conjugate symmetric forms
7 Related products
8 See also
9 Notes
10 References
11 Sources
Definition[edit]
In this article, the field of scalars denoted F is either the field of real numbers R or the field of complex numbers C.
Formally, an inner product space is a vector space V over the field F together with an inner product, i.e., with a map
⟨ ⋅ , ⋅ ⟩ : V × V → F {\displaystyle \langle \cdot ,\cdot \rangle :V\times V\to F}
that satisfies the following three axioms for all vectors x, y, z ∈ V and all scalars a ∈ F:[2][3]
Conjugate symmetry:[Note 1]
⟨ x , y ⟩ = ⟨ y , x ⟩ ¯ {\displaystyle \langle x,y\rangle ={\overline {\langle y,x\rangle }}}
Linearity in the first argument:
⟨ a x , y ⟩ = a ⟨ x , y ⟩ ⟨ x + y , z ⟩ = ⟨ x , z ⟩ + ⟨ y , z ⟩ {\displaystyle {\begin{aligned}\langle ax,y\rangle &=a\langle x,y\rangle \\\langle x+y,z\rangle &=\langle x,z\rangle +\langle y,z\rangle \end{aligned}}}
Positive-definiteness:
⟨ x , x ⟩ ≥ 0 ⟨ x , x ⟩ = 0 ⇔ x = 0 . {\displaystyle {\begin{aligned}\langle x,x\rangle &\geq 0\\\langle x,x\rangle &=0\Leftrightarrow x=\mathbf {0} \,.\end{aligned}}}
Alternative definitions, notations and remarks[edit]
Some authors, especially in physics and matrix algebra, prefer to define the inner product and the sesquilinear form with linearity in the second argument rather than the first. Then the first argument becomes conjugate linear, rather than the second. In those disciplines we would write the product ⟨x,y⟩ as ⟨y|x⟩ (the bra–ket notation of quantum mechanics), respectively y†x (dot product as a case of the convention of forming the matrix product AB as the dot products of rows of A with columns of B). Here the kets and columns are identified with the vectors of V and the bras and rows with the linear functionals (covectors) of the dual space V∗, with conjugacy associated with duality. This reverse order is now occasionally followed in the more abstract literature,[4] taking ⟨x,y⟩ to be conjugate linear in x rather than y. A few instead find a middle ground by recognizing both ⟨·,·⟩ and ⟨·|·⟩ as distinct notations differing only in which argument is conjugate linear.
There are various technical reasons why it is necessary to restrict the basefield to R and C in the definition. Briefly, the basefield has to contain an ordered subfield in order for non-negativity to make sense,[5] and therefore has to have characteristic equal to 0 (since any ordered field has to have such characteristic). This immediately excludes finite fields. The basefield has to have additional structure, such as a distinguished automorphism. More generally any quadratically closed subfield of R or C will suffice for this purpose, e.g., the algebraic numbers or the constructible numbers. However in these cases when it is a proper subfield (i.e., neither R nor C) even finite-dimensional inner product spaces will fail to be metrically complete. In contrast all finite-dimensional inner product spaces over R or C, such as those used in quantum computation, are automatically metrically complete and hence Hilbert spaces.
In some cases we need to consider non-negative semi-definite sesquilinear forms. This means that ⟨x,x⟩ is only required to be non-negative. We show how to treat these below.
Elementary properties[edit]
When F = R, conjugate symmetry reduces to symmetry. That is, ⟨x,y⟩ = ⟨y,x⟩ for F = R; while for F = C, ⟨x,y⟩ is equal to the complex conjugate.
Notice that conjugate symmetry implies that ⟨x,x⟩ is real for all x, since we have:
⟨ x , x ⟩ = ⟨ x , x ⟩ ¯ . {\displaystyle \langle x,x\rangle ={\overline {\langle x,x\rangle }}\,.}
Moreover, sesquilinearity (see below) implies that
⟨ − x , x ⟩ = − 1 ⟨ x , x ⟩ = − 1 ¯ ⟨ x , x ⟩ = ⟨ x , − x ⟩ . {\displaystyle \langle -x,x\rangle =-1\langle x,x\rangle ={\overline {-1}}\langle x,x\rangle =\langle x,-x\rangle \,.}
Conjugate symmetry and linearity in the first variable gives
⟨ x , a y ⟩ = ⟨ a y , x ⟩ ¯ = a ¯ ⟨ y , x ⟩ ¯ = a ¯ ⟨ x , y ⟩ ⟨ x , y + z ⟩ = ⟨ y + z , x ⟩ ¯ = ⟨ y , x ⟩ ¯ + ⟨ z , x ⟩ ¯ = ⟨ x , y ⟩ + ⟨ x , z ⟩ , {\displaystyle {\begin{aligned}\langle x,ay\rangle &={\overline {\langle ay,x\rangle }}={\overline {a}}{\overline {\langle y,x\rangle }}={\overline {a}}\langle x,y\rangle \\\langle x,y+z\rangle &={\overline {\langle y+z,x\rangle }}={\overline {\langle y,x\rangle }}+{\overline {\langle z,x\rangle }}=\langle x,y\rangle +\langle x,z\rangle \,,\end{aligned}}}
so an inner product is a sesquilinear form. Conjugate symmetry is also called Hermitian symmetry, and a conjugate symmetric sesquilinear form is called a Hermitian form. While the above axioms are more mathematically economical, a compact verbal definition of an inner product is a positive-definite Hermitian form.
In the case of F = R, conjugate-symmetry reduces to symmetry, and sesquilinear reduces to bilinear. So, an inner product on a real vector space is a positive-definite symmetric bilinear form.
From the linearity property it is derived that x = 0 implies ⟨x,x⟩ = 0. while from the positive-definiteness axiom we obtain the converse, ⟨x,x⟩ = 0 implies x = 0. Combining these two, we have the property that ⟨x,x⟩ = 0 if and only if x = 0.
Combining the linearity of the inner product in its first argument and the conjugate symmetry gives the following important generalization of the familiar square expansion:
⟨ x + y , x + y ⟩ = ⟨ x , x ⟩ + ⟨ x , y ⟩ + ⟨ y , x ⟩ + ⟨ y , y ⟩ . {\displaystyle \langle x+y,x+y\rangle =\langle x,x\rangle +\langle x,y\rangle +\langle y,x\rangle +\langle y,y\rangle \,.}
Assuming the underlying field to be R, the inner product becomes symmetric, and we obtain
⟨ x ± y , x ± y ⟩ = ⟨ x , x ⟩ ± 2 ⟨ x , y ⟩ + ⟨ y , y ⟩ , {\displaystyle \langle x\pm y,x\pm y\rangle =\langle x,x\rangle \pm 2\langle x,y\rangle +\langle y,y\rangle \,,}
The property of an inner product space V that
⟨ x + y , z ⟩ = ⟨ x , z ⟩ + ⟨ y , z ⟩ , ⟨ x , y + z ⟩ = ⟨ x , y ⟩ + ⟨ x , z ⟩ {\displaystyle {\begin{aligned}\langle x+y,z\rangle &=\langle x,z\rangle +\langle y,z\rangle \,,\\\langle x,y+z\rangle &=\langle x,y\rangle +\langle x,z\rangle \end{aligned}}}
is also known as additivity.
Examples[edit]
Real numbers[edit]
A simple example is the real numbers with the standard multiplication as the inner product
⟨ x , y ⟩ := x y . {\displaystyle \langle x,y\rangle :=xy.}
Euclidean space[edit]
More generally, the real n-space Rn with the dot product is an inner product space, an example of a Euclidean n-space.
⟨ [ x 1 ⋮ x n ] , [ y 1 ⋮ y n ] ⟩ := x T y = ∑ i = 1 n x i y i = x 1 y 1 + ⋯ + x n y n , {\displaystyle \left\langle {\begin{bmatrix}x_{1}\\\vdots \\x_{n}\end{bmatrix}},{\begin{bmatrix}y_{1}\\\vdots \\y_{n}\end{bmatrix}}\right\rangle :=x^{\mathsf {T}}y=\sum _{i=1}^{n}x_{i}y_{i}=x_{1}y_{1}+\cdots +x_{n}y_{n},}
where xT is the transpose of x.
Complex coordinate space[edit]
The general form of an inner product on Cn is known as the Hermitian form and is given by
⟨ x , y ⟩ := y † M x = x † M y ¯ , {\displaystyle \langle \mathbf {x} ,\mathbf {y} \rangle :=\mathbf {y} ^{\dagger }\mathbf {M} \mathbf {x} ={\overline {\mathbf {x} ^{\dagger }\mathbf {M} \mathbf {y} }},}
where M is any Hermitian positive-definite matrix and y† is the conjugate transpose of y. For the real case this corresponds to the dot product of the results of directionally different scaling of the two vectors, with positive scale factors and orthogonal directions of scaling. Up to an orthogonal transformation it is a weighted-sum version of the dot product, with positive weights.
Hilbert space[edit]
The article on Hilbert space has several examples of inner product spaces wherein the metric induced by the inner product yields a complete metric space. An example of an inner product which induces an incomplete metric occurs with the space C([a,b]) of continuous complex valued functions on the interval [a,b]. The inner product is
⟨ f , g ⟩ := ∫ a b f ( t ) g ( t ) ¯ d t . {\displaystyle \langle f,g\rangle :=\int _{a}^{b}f(t){\overline {g(t)}}\,\mathrm {d} t.}
This space is not complete; consider for example, for the interval [−1,1] the sequence of continuous "step" functions, { fk}k, defined by:
f k ( t ) = { 0 t ∈ [ − 1 , 0 ] 1 t ∈ [ 1 k , 1 ] k t t ∈ ( 0 , 1 k ) {\displaystyle f_{k}(t)={\begin{cases}0&t\in [-1,0]\\1&t\in \left[{\tfrac {1}{k}},1\right]\\kt&t\in \left(0,{\tfrac {1}{k}}\right)\end{cases}}}
This sequence is a Cauchy sequence for the norm induced by the preceding inner product, which does not converge to a continuous function.
Random variables[edit]
For real random variables X and Y, the expected value of their product
⟨ X , Y ⟩ := E  ( X Y ) {\displaystyle \langle X,Y\rangle :=\operatorname {E} (XY)}
is an inner product.[6][7][8] In this case, ⟨X,X⟩ = 0 if and only if Pr(X = 0) = 1 (i.e., X = 0 almost surely). This definition of expectation as inner product can be extended to random vectors as well.
Real matrices[edit]
For real matrices of the same size, ⟨A,B⟩ := tr(ABT) with transpose as conjugation
( ⟨ A , B ⟩ = ⟨ B T , A T ⟩ ) {\displaystyle \left(\langle A,B\rangle =\left\langle B^{\mathrm {T} },A^{\mathrm {T} }\right\rangle \right)}
is an inner product.
Vector spaces with forms[edit]
On an inner product space, or more generally a vector space with a nondegenerate form (so an isomorphism V → V∗) vectors can be sent to covectors (in coordinates, via transpose), so one can take the inner product and outer product of two vectors, not simply of a vector and a covector.
Norms on inner product spaces[edit]
A linear space with a norm such as:
‖ x ‖ p = ( ∑ i = 1 ∞ | ξ i | p ) 1 p x = { ξ i } ∈ l p , p ≠ 2 , {\displaystyle \|x\|_{p}=\left(\sum _{i=1}^{\infty }\left|\xi _{i}\right|^{p}\right)^{\frac {1}{p}}\qquad x=\left\{\xi _{i}\right\}\in l^{p},\quad p\neq 2,}
is a normed space but not an inner product space, because this norm does not satisfy the parallelogram equality required of a norm to have an inner product associated with it.[9][10]
However, inner product spaces have a naturally defined norm based upon the inner product of the space itself that does satisfy the parallelogram equality:
‖ x ‖ = ⟨ x , x ⟩ . {\displaystyle \|x\|={\sqrt {\langle x,x\rangle }}.}
This is well defined by the nonnegativity axiom of the definition of inner product space. The norm is thought of as the length of the vector x. Directly from the axioms, we can prove the following:
Cauchy–Schwarz inequality: for x, y elements of V
| ⟨ x , y ⟩ | ≤ ‖ x ‖ ⋅ ‖ y ‖ {\displaystyle |\langle x,y\rangle |\leq \|x\|\cdot \|y\|}
with equality if and only if x and y are linearly dependent. This is one of the most important inequalities in mathematics. It is also known in the Russian mathematical literature as the Cauchy–Bunyakovsky–Schwarz inequality.
Orthogonality: The geometric interpretation of the inner product in terms of angle and length, motivates much of the geometric terminology we use in regard to these spaces. Indeed, an immediate consequence of the Cauchy–Schwarz inequality is that it justifies defining the angle between two non-zero vectors x and y (denoted ∠) in the case F = R by the identity
∠ ( x , y ) = arccos  ⟨ x , y ⟩ ‖ x ‖ ⋅ ‖ y ‖ . {\displaystyle \angle (x,y)=\arccos {\frac {\langle x,y\rangle }{\|x\|\cdot \|y\|}}.}
We assume the value of the angle is chosen to be in the interval [0, π]. This is in analogy to the situation in two-dimensional Euclidean space.
In the case F = C, the angle in the interval [0, π/2] is typically defined by
∠ ( x , y ) = arccos  | ⟨ x , y ⟩ | ‖ x ‖ ⋅ ‖ y ‖ . {\displaystyle \angle (x,y)=\arccos {\frac {|\langle x,y\rangle |}{\|x\|\cdot \|y\|}}.}
Correspondingly, we will say that non-zero vectors x and y of V are orthogonal if and only if their inner product is zero.
Homogeneity: for x an element of V and r a scalar
‖ r ⋅ x ‖ = | r | ⋅ ‖ x ‖ . {\displaystyle \|r\cdot x\|=|r|\cdot \|x\|.}
The homogeneity property is completely trivial to prove.
Triangle inequality: for x, y elements of V
‖ x + y ‖ ≤ ‖ x ‖ + ‖ y ‖ . {\displaystyle \|x+y\|\leq \|x\|+\|y\|.}
The last two properties show the function defined is indeed a norm.
Because of the triangle inequality and because of axiom 2, we see that ||·|| is a norm which turns V into a normed vector space and hence also into a metric space. The most important inner product spaces are the ones which are complete with respect to this metric; they are called Hilbert spaces. Every inner product V space is a dense subspace of some Hilbert space. This Hilbert space is essentially uniquely determined by V and is constructed by completing V.
Pythagorean theorem: Whenever x, y are in V and ⟨x,y⟩ = 0, then
‖ x ‖ 2 + ‖ y ‖ 2 = ‖ x + y ‖ 2 . {\displaystyle \|x\|^{2}+\|y\|^{2}=\|x+y\|^{2}.}
The proof of the identity requires only expressing the definition of norm in terms of the inner product and multiplying out, using the property of additivity of each component.
The name Pythagorean theorem arises from the geometric interpretation of this result as an analogue of the theorem in synthetic geometry. Note that the proof of the Pythagorean theorem in synthetic geometry is considerably more elaborate because of the paucity of underlying structure. In this sense, the synthetic Pythagorean theorem, if correctly demonstrated, is deeper than the version given above.
An induction on the Pythagorean theorem yields:
If x1, ..., xn are orthogonal vectors, that is, ⟨xj,xk⟩ = 0 for distinct indices j, k, then
∑ i = 1 n ‖ x i ‖ 2 = ‖ ∑ i = 1 n x i ‖ 2 . {\displaystyle \sum _{i=1}^{n}\|x_{i}\|^{2}=\left\|\sum _{i=1}^{n}x_{i}\right\|^{2}.}
In view of the Cauchy-Schwarz inequality, we also note that ⟨·,·⟩ is continuous from V × V to F. This allows us to extend Pythagoras' theorem to infinitely many summands:
Parseval's identity: Suppose V is a complete inner product space. If {xk} are mutually orthogonal vectors in V then
∑ i = 1 ∞ ‖ x i ‖ 2 = ‖ ∑ i = 1 ∞ x i ‖ 2 , {\displaystyle \sum _{i=1}^{\infty }\|x_{i}\|^{2}=\left\|\sum _{i=1}^{\infty }x_{i}\right\|^{2},}
provided the infinite series on the left is convergent. Completeness of the space is needed to ensure that the sequence of partial sums
S k = ∑ i = 1 k x i , {\displaystyle S_{k}=\sum _{i=1}^{k}x_{i},}
which is easily shown to be a Cauchy sequence, is convergent.
Parallelogram law: for x, y elements of V,
‖ x + y ‖ 2 + ‖ x − y ‖ 2 = 2 ‖ x ‖ 2 + 2 ‖ y ‖ 2 . {\displaystyle \|x+y\|^{2}+\|x-y\|^{2}=2\|x\|^{2}+2\|y\|^{2}.}
The Parallelogram law is, in fact, a necessary and sufficient condition for the existence of a scalar product corresponding to a given norm. If it holds, the scalar product is defined by the polarization identity:
‖ x + y ‖ 2 = ‖ x ‖ 2 + ‖ y ‖ 2 + 2 ℜ ⟨ x , y ⟩ . {\displaystyle \|x+y\|^{2}=\|x\|^{2}+\|y\|^{2}+2\Re \langle x,y\rangle .}
which is a form of the law of cosines.
Orthonormal sequences[edit]
Let V be a finite dimensional inner product space of dimension n. Recall that every basis of V consists of exactly n linearly independent vectors. Using the Gram–Schmidt process we may start with an arbitrary basis and transform it into an orthonormal basis. That is, into a basis in which all the elements are orthogonal and have unit norm. In symbols, a basis {e1, ..., en} is orthonormal if ⟨ei,ej⟩ = 0 for every i ≠ j and ⟨ei,ei⟩ = ||ei|| = 1 for each i.
This definition of orthonormal basis generalizes to the case of infinite-dimensional inner product spaces in the following way. Let V be any inner product space. Then a collection
E = { e α } α ∈ A {\displaystyle E=\left\{e_{\alpha }\right\}_{\alpha \in A}}
is a basis for V if the subspace of V generated by finite linear combinations of elements of E is dense in V (in the norm induced by the inner product). We say that E is an orthonormal basis for V if it is a basis and
⟨ e α , e β ⟩ = 0 {\displaystyle \left\langle e_{\alpha },e_{\beta }\right\rangle =0}
if α ≠ β and ⟨eα,eα⟩ = ||eα|| = 1 for all α, β ∈ A.
Using an infinite-dimensional analog of the Gram-Schmidt process one may show:
Theorem. Any separable inner product space V has an orthonormal basis.
Using the Hausdorff maximal principle and the fact that in a complete inner product space orthogonal projection onto linear subspaces is well-defined, one may also show that
Theorem. Any complete inner product space V has an orthonormal basis.
The two previous theorems raise the question of whether all inner product spaces have an orthonormal basis. The answer, it turns out is negative. This is a non-trivial result, and is proved below. The following proof is taken from Halmos's A Hilbert Space Problem Book (see the references).[citation needed]
showProof
Recall that the dimension of an inner product space is the cardinality of a maximal orthonormal system that it contains (by Zorn's lemma it contains at least one, and any two have the same cardinality). An orthonormal basis is certainly a maximal orthonormal system, but as we shall see, the converse need not hold. Observe that if G is a dense subspace of an inner product space H, then any orthonormal basis for G is automatically an orthonormal basis for H. Thus, it suffices to construct an inner product space H with a dense subspace G whose dimension is strictly smaller than that of H.
Let K be a Hilbert space of dimension ℵ0 (for instance, K = l2(N)). Let E be an orthonormal basis of K, so |E| = ℵ0. Extend E to a Hamel basis E ∪ F for K, where E ∩ F = ∅. Since it is known that the Hamel dimension of K is c, the cardinality of the continuum, it must be that |F| = c.
Let L be a Hilbert space of dimension c (for instance, L = l2(R)). Let B be an orthonormal basis for L, and let φ : F → B be a bijection. Then there is a linear transformation T : K → L such that Tf = φ( f ) for f ∈ F, and Te = 0 for e ∈ E.
Let H = K ⊕ L and let G = {(k,Tk) : k ∈ K)} be the graph of T. Let Ḡ be the closure of G in H; we will show Ḡ = H. Since for any e ∈ E we have (e,0) ∈ G, it follows that K ⊕ 0 ⊂ Ḡ.
Next, if b ∈ B, then b = Tf for some f ∈ F ⊂ K, so ( f,b) ∈ G ⊂ Ḡ; since ( f,0) ∈ Ḡ as well, we also have (0,b) ∈ Ḡ. It follows that 0 ⊕ L ⊂ Ḡ, so Ḡ = H, and G is dense in H.
Finally, {(e,0) : e ∈ E} is a maximal orthonormal set in G; if
0 = ⟨ ( e , 0 ) , ( k , T k ) ⟩ = ⟨ e , k ⟩ + ⟨ 0 , T k ⟩ = ⟨ e , k ⟩ {\displaystyle 0=\langle (e,0),(k,Tk)\rangle =\langle e,k\rangle +\langle 0,Tk\rangle =\langle e,k\rangle }
for all e ∈ E then certainly k = 0, so (k,Tk) = (0,0) is the zero vector in G. Hence the dimension of G is |E| = ℵ0, whereas it is clear that the dimension of H is c. This completes the proof.
Parseval's identity leads immediately to the following theorem:
Theorem. Let V be a separable inner product space and {ek}k an orthonormal basis of V. Then the map
x ↦ { ⟨ e k , x ⟩ } k ∈ N {\displaystyle x\mapsto {\bigl \{}\langle e_{k},x\rangle {\bigr \}}_{k\in \mathbb {N} }}
is an isometric linear map V → l2 with a dense image.
This theorem can be regarded as an abstract form of Fourier series, in which an arbitrary orthonormal basis plays the role of the sequence of trigonometric polynomials. Note that the underlying index set can be taken to be any countable set (and in fact any set whatsoever, provided l2 is defined appropriately, as is explained in the article Hilbert space). In particular, we obtain the following result in the theory of Fourier series:
Theorem. Let V be the inner product space C[−π,π]. Then the sequence (indexed on set of all integers) of continuous functions
e k ( t ) = e i k t 2 π {\displaystyle e_{k}(t)={\frac {e^{ikt}}{\sqrt {2\pi }}}}
is an orthonormal basis of the space C[−π,π] with the L2 inner product. The mapping
f ↦ 1 2 π { ∫ − π π f ( t ) e − i k t d t } k ∈ Z {\displaystyle f\mapsto {\frac {1}{\sqrt {2\pi }}}\left\{\int _{-\pi }^{\pi }f(t)e^{-ikt}\,\mathrm {d} t\right\}_{k\in \mathbb {Z} }}
is an isometric linear map with dense image.
Orthogonality of the sequence {ek}k follows immediately from the fact that if k ≠ j, then
∫ − π π e − i ( j − k ) t d t = 0. {\displaystyle \int _{-\pi }^{\pi }e^{-i(j-k)t}\,\mathrm {d} t=0.}
Normality of the sequence is by design, that is, the coefficients are so chosen so that the norm comes out to 1. Finally the fact that the sequence has a dense algebraic span, in the inner product norm, follows from the fact that the sequence has a dense algebraic span, this time in the space of continuous periodic functions on [−π,π] with the uniform norm. This is the content of the Weierstrass theorem on the uniform density of trigonometric polynomials.
Operators on inner product spaces[edit]
Several types of linear maps A from an inner product space V to an inner product space W are of relevance:
Continuous linear maps, i.e., A is linear and continuous with respect to the metric defined above, or equivalently, A is linear and the set of non-negative reals {||Ax||}, where x ranges over the closed unit ball of V, is bounded.
Symmetric linear operators, i.e., A is linear and ⟨Ax,y⟩ = ⟨x,Ay⟩ for all x, y in V.
Isometries, i.e., A is linear and ⟨Ax,Ay⟩ = ⟨x,y⟩ for all x, y in V, or equivalently, A is linear and ||Ax|| = ||x|| for all x in V. All isometries are injective. Isometries are morphisms between inner product spaces, and morphisms of real inner product spaces are orthogonal transformations (compare with orthogonal matrix).
Isometrical isomorphisms, i.e., A is an isometry which is surjective (and hence bijective). Isometrical isomorphisms are also known as unitary operators (compare with unitary matrix).
From the point of view of inner product space theory, there is no need to distinguish between two spaces which are isometrically isomorphic. The spectral theorem provides a canonical form for symmetric, unitary and more generally normal operators on finite dimensional inner product spaces. A generalization of the spectral theorem holds for continuous normal operators in Hilbert spaces.
Generalizations[edit]
Any of the axioms of an inner product may be weakened, yielding generalized notions. The generalizations that are closest to inner products occur where bilinearity and conjugate symmetry are retained, but positive-definiteness is weakened.
Degenerate inner products[edit]
If V is a vector space and ⟨·,···⟩ a semi-definite sesquilinear form, then the function:
‖ x ‖ = ⟨ x , x ⟩ {\displaystyle \|x\|={\sqrt {\langle x,x\rangle }}}
makes sense and satisfies all the properties of norm except that ||x|| = 0 does not imply x = 0 (such a functional is then called a semi-norm). We can produce an inner product space by considering the quotient W = V/{x : ||x|| = 0}. The sesquilinear form ⟨·,·⟩ factors through W.
This construction is used in numerous contexts. The Gelfand–Naimark–Segal construction is a particularly important example of the use of this technique. Another example is the representation of semi-definite kernels on arbitrary sets.
Nondegenerate conjugate symmetric forms[edit]
Main article: Pseudo-Euclidean space
Alternatively, one may require that the pairing be a nondegenerate form, meaning that for all non-zero x there exists some y such that ⟨x,y⟩ ≠ 0, though y need not equal x; in other words, the induced map to the dual space V → V∗ is injective. This generalization is important in differential geometry: a manifold whose tangent spaces have an inner product is a Riemannian manifold, while if this is related to nondegenerate conjugate symmetric form the manifold is a pseudo-Riemannian manifold. By Sylvester's law of inertia, just as every inner product is similar to the dot product with positive weights on a set of vectors, every nondegenerate conjugate symmetric form is similar to the dot product with nonzero weights on a set of vectors, and the number of positive and negative weights are called respectively the positive index and negative index. Product of vectors in Minkowski space is an example of indefinite inner product, although, technically speaking, it is not an inner product according to the standard definition above. Minkowski space has four dimensions and indices 3 and 1 (assignment of "+" and "−" to them differs depending on conventions).
Purely algebraic statements (ones that do not use positivity) usually only rely on the nondegeneracy (the injective homomorphism V → V∗) and thus hold more generally.
Related products[edit]
The term "inner product" is opposed to outer product, which is a slightly more general opposite. Simply, in coordinates, the inner product is the product of a 1 × n covector with an n × 1 vector, yielding a 1 × 1 matrix (a scalar), while the outer product is the product of an m × 1 vector with a 1 × n covector, yielding an m × n matrix. Note that the outer product is defined for different dimensions, while the inner product requires the same dimension. If the dimensions are the same, then the inner product is the trace of the outer product (trace only being properly defined for square matrices). In a quip: "inner is horizontal times vertical and shrinks down, outer is vertical times horizontal and expands out".
More abstractly, the outer product is the bilinear map W × V∗ → Hom(V,W) sending a vector and a covector to a rank 1 linear transformation (simple tensor of type (1,1)), while the inner product is the bilinear evaluation map V∗ × V → F given by evaluating a covector on a vector; the order of the domain vector spaces here reflects the covector/vector distinction.
The inner product and outer product should not be confused with the interior product and exterior product, which are instead operations on vector fields and differential forms, or more generally on the exterior algebra.
As a further complication, in geometric algebra the inner product and the exterior (Grassmann) product are combined in the geometric product (the Clifford product in a Clifford algebra) – the inner product sends two vectors (1-vectors) to a scalar (a 0-vector), while the exterior product sends two vectors to a bivector (2-vector) – and in this context the exterior product is usually called the "outer (alternatively, wedge) product". The inner product is more correctly called a scalar product in this context, as the nondegenerate quadratic form in question need not be positive definite (need not be an inner product).
See also[edit]
Space (mathematics)
Normed vector space
Energetic space
Dual space
Biorthogonal system
Bilinear form
Notes[edit]
Jump up ^ A bar over an expression denotes complex conjugation.
References[edit]
Jump up ^ Moore, Gregory H. (1995). "The axiomatization of linear algebra: 1875-1940". Historia Mathematica. 22 (3): 262–303. doi:10.1006/hmat.1995.1025.
Jump up ^ Jain, P. K.; Ahmad, Khalil (1995). "5.1 Definitions and basic properties of inner product spaces and Hilbert spaces". Functional Analysis (2nd ed.). New Age International. p. 203. ISBN 81-224-0801-X.
Jump up ^ Prugovec̆ki, Eduard (1981). "Definition 2.1". Quantum Mechanics in Hilbert Space (2nd ed.). Academic Press. pp. 18ff. ISBN 0-12-566060-X.
Jump up ^ Emch, Gerard G. (1972). Algebraic Methods in Statistical Mechanics and Quantum Field Theory. New York: Wiley-Interscience. ISBN 978-0-471-23900-0.
Jump up ^ Finkbeiner, Daniel T. (2013), Introduction to Matrices and Linear Transformations, Dover Books on Mathematics (3rd ed.), Courier Dover Publications, p. 242, ISBN 9780486279664 .
Jump up ^ Ouwehand, Peter (November 2010). "Spaces of Random Variables" (PDF). AIMS. Retrieved 2017-09-05.
Jump up ^ Siegrist, Kyle (1997). "Vector Spaces of Random Variables". Random: Probability, Mathematical Statistics, Stochastic Processes. Retrieved 2017-09-05.
Jump up ^ Bigoni, Daniele (2015). "Appendix B: Probability theory and functional spaces". Uncertainty Quantification with Applications to Engineering Problems (PDF) (PhD). Technical University of Denmark. Retrieved 2017-09-05.
Jump up ^ Jain, P. K.; Ahmad, Khalil (1995). "Example 5". Functional Analysis (2nd ed.). New Age International. p. 209. ISBN 81-224-0801-X.
Jump up ^ Saxe, Karen (2002). Beginning Functional Analysis. Springer. p. 7. ISBN 0-387-95224-1.
Sources[edit]
Axler, Sheldon (1997). Linear Algebra Done Right (2nd ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-387-98258-8.
Emch, Gerard G. (1972). Algebraic Methods in Statistical Mechanics and Quantum Field Theory. Wiley-Interscience. ISBN 978-0-471-23900-0.
Young, Nicholas (1988). An Introduction to Hilbert Space. Cambridge University Press. ISBN 978-0-521-33717-5.

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values: מרחב מכפלה פנימית – Hebrew

values:
interlanguage-link interwiki-he
display: none;

values:
interlanguage-link interwiki-nl
display: none;

values: 計量ベクトル空間 – Japanese

values:
interlanguage-link interwiki-nn
display: none;

values:
interlanguage-link interwiki-pl
display: none;

values:
interlanguage-link interwiki-ro
display: none;

values: Предгильбертово пространство – Russian

values:
interlanguage-link interwiki-sk
display: none;

values:
interlanguage-link interwiki-sl
display: none;

values:
interlanguage-link interwiki-fi
display: none;

values:
interlanguage-link interwiki-sv
display: none;

values: اندرونی حاصل ضرب فضا – Urdu

values: All languages (initial selection from common choices by you and others)

values: /static/images/wikimedia-button-1.5x.png 1.5x, /static/images/wikimedia-button-2x.png 2x

values: /static/images/poweredby_mediawiki_132x47.png 1.5x, /static/images/poweredby_mediawiki_176x62.png 2x

values:
suggestions
display: none; font-size: 13px;

values: mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-Inner_product_space rootpage-Inner_product_space skin-vector action-view