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Burt Lancaster

Burton Stephen Lancaster (November 2, 1913 – October 20, 1994) was an American actor and film producer. Initially known for playing tough guys with a tender heart, he went on to achieve success with more complex and challenging roles over a 45-year career in films and television series. He was a four-time nominee for the Academy Award for Best Actor (winning once), and he also won two BAFTA Awards and one Golden Globe Award for Best Lead Actor. The American Film Institute ranks Lancaster as # of the greatest male stars of classic Hollywood cinema.

Lancaster performed as a circus acrobat in the 1930s. At the age of 32, and after serving in World War II, he landed a role in a Broadway play and drew the attention of a Hollywood agent. His breakthrough role was in the film noir The Killers in 1946 alongside Ava Gardner. A critical success, it launched both of their careers. Not long after in 1948, Lancaster starred alongside Barbara Stanwyck in the commercially and critically acclaimed film Sorry, Wrong Number where he portrayed the husband to her bedridden, invalid character. In 1953, Lancaster played the illicit lover of Deborah Kerr in the military drama From Here to Eternity. A box office smash, it won eight Academy Awards, including Best Picture, and landed a Best Actor nomination for Lancaster.

Later in the 1950s, he starred in The Rainmaker (1956), with Katharine Hepburn, earning a Best Actor Golden Globe nomination, and in 1957 he starred in Gunfight at the O.K. Corral (1957) with frequent co-star Kirk Douglas. During the 1950s, his production company, Hecht-Hill-Lancaster, was highly successful, with Lancaster acting in films such as: Trapeze (1956), a box office smash in which he used his acrobatic skills and for which he won the Silver Bear for Best Actor; Sweet Smell of Success (1957), a dark drama today considered a classic; Run Silent, Run Deep (1958), a WWII submarine drama with Clark Gable; and Separate Tables (1958), a hotel-set drama which received seven Oscar nominations.

In the early 1960s, Lancaster starred in a string of critically successful films, each in very disparate roles. Playing a charismatic biblical con-man in Elmer Gantry in 1960 won him the Academy Award and the Golden Globe for Best Actor. He played a Nazi war criminal in 1961 in the all-star, war-crime-trial film, Judgment at Nuremberg. Playing a bird expert prisoner in Birdman of Alcatraz in 1962, he earned the BAFTA Award for Best Foreign Actor and his third Oscar nomination. In 1963, Lancaster traveled to Italy to star as an Italian prince in Visconti's epic period drama The Leopard. In 1964, he played a US Air Force General who, opposed by a Colonel played by Douglas, tries to overthrow the President in Seven Days in May. Then, in 1966, he played an explosives expert in the western The Professionals. Although the reception of his 1968 film The Swimmer was initially lackluster upon release, in the years after it has grown in stature critically and attained a cult following.

In 1970, Lancaster starred in the box-office hit, air-disaster drama Airport. In 1974 he again starred in a Visconti film, Conversation Piece. He experienced a career resurgence in 1980 with the crime-romance Atlantic City, winning the BAFTA for Best Actor and landing his fourth Oscar nomination. Starting in the late 1970s, he also appeared in television mini-series, including the award-winning Separate but Equal with Sidney Poitier. He continued acting into his late 70s, until a stroke in 1990 forced him to retire; four years later he died from a heart attack. His final film role was in the Oscar-nominated Field of Dreams.

Early life
Lancaster was born on November 2, 1913, in New York City, at his parents' home at 209 East 106th Street, the son of Elizabeth (née Roberts) and mailman James Lancaster. Both of his parents were Protestants of working-class origin. All four of his grandparents were emigrants from Ireland to the United States, from the province of Ulster. His maternal grandparents were from Belfast and were descendants of English dissenters who had colonised Ireland as part of the Plantation of Ulster.

Lancaster grew up in East Harlem, New York City. He developed a great interest and skill in gymnastics while attending DeWitt Clinton High School, where he was a basketball star. Before he graduated from DeWitt Clinton, his mother died of a cerebral hemorrhage. Lancaster was accepted by New York University with an athletic scholarship, but dropped out.

Circus career
At the age of 9, Lancaster met Nick Cravat with whom he developed a lifelong partnership. Together, they learned to act in local theatre productions and circus arts at Union Settlement, one of the city's oldest settlement houses. In the 1930s, they formed the acrobat duo Lang and Cravat and soon joined the Kay Brothers circus. However, in 1939, an injury forced Lancaster to give up the profession, with great regret. He then found temporary work, first as a salesman for Marshall Fields and then as a singing waiter in various restaurants.

World War II service
After the United States entered World War II, Lancaster joined the United States Army in January 1943 and performed with the Army's 21st Special Services Division, one of the military groups organized to follow the troops on the ground and provide USO entertainment to keep up morale. He served with General Mark Clark's Fifth Army in Italy from 1943 to 1945. He was discharged October 1945 and was an entertainment specialist with the rank of technician fifth grade.

Broadway
Lancaster returned to New York after his Army service. Although initially unenthusiastic about acting, Lancaster was encouraged to audition for a Broadway play by a producer who saw him in an elevator while he was visiting his then-girlfriend at work. The audition was successful and Lancaster was cast in Harry Brown's A Sound of Hunting (1945). The show only ran three weeks, but his performance attracted the interest of a Hollywood agent, Harold Hecht. Lancaster had other offers but Hecht promised him the opportunity to produce their own movies within five years of hitting Hollywood.

Through Hecht, Lancaster was brought to the attention of producer Hal B. Wallis. Lancaster left New York and moved to Los Angeles. Wallis signed him to a non-exclusive eight-movie contract.

Hal Wallis
Lancaster's first filmed movie was Desert Fury for Wallis in 1947, where Lancaster was billed after John Hodiak and Lizabeth Scott. It was directed by Lewis Allen.

Then producer Mark Hellinger approached him to star in 1946's The Killers, which was completed and released prior to Desert Fury. Directed by Robert Siodmak, it was a great commercial and critical success and launched Lancaster and his co-star Ava Gardner to stardom. It has since come to be regarded as a classic.

Hellinger used Lancaster again on Brute Force in 1947, a prison drama written by Richard Brooks and directed by Jules Dassin. It was also well received. Wallis released his films through Paramount, and so Lancaster and other Wallis contractees made cameos in Variety Girl in 1947.

Lancaster's next film was a thriller for Wallis in 1947, I Walk Alone, co-starring Lizabeth Scott and a young Kirk Douglas, who was also under contract to Wallis. Variety listed it as one of the top grossers of the year, taking in more than $2 million.

In 1948, Lancaster had a change of pace with the film adaptation of Arthur Miller's All My Sons, made at Universal Pictures with Edward G. Robinson. His third film for Wallis was an adaptation of Sorry, Wrong Number in 1948, with Barbara Stanwyck.

Norma Productions
Hecht kept to his promise to Lancaster to turn producer. The two of them formed a company, Norma Productions, and did a deal with Universal to make a thriller about a disturbed G.I. in London, Kiss the Blood Off My Hands in 1948, with Joan Fontaine and directed by Norman Foster. It made a profit of only $50,000, but was critically acclaimed.

Back in Hollywood, Lancaster made another film noir with Siodmak, Criss Cross, in 1949. It was originally going to be produced by Hellinger and when Hellinger died, another took over. Tony Curtis made an early appearance.

Lancaster appeared in a fourth picture for Wallis, Rope of Sand, in 1949.

Norma Productions signed a three-picture deal with Warner Bros. The first was 1950's The Flame and the Arrow, a swashbuckler movie, in which Lancaster drew on his circus skills. Nick Cravat had a supporting role and the film was a huge commercial success, making $6 million. It was Warners' most popular film of the year and established an entirely new image for Lancaster.

Lancaster was borrowed by 20th Century Fox for Mister 880 in 1950, a comedy with Edmund Gwenn. MGM put him in a popular Western, Vengeance Valley in 1951, then he went to Warners to play the title role in the biopic Jim Thorpe – All-American, also in 1951.

Halburt
Norma signed a deal with Columbia Pictures to make two films through a Norma subsidiary, Halburt. The first film was 1951's Ten Tall Men, where Lancaster was a member of the French Foreign Legion. Robert Aldrich worked on the movie as a production manager.

The second was 1952's The First Time, a comedy which was the directorial debut of Frank Tashlin. It was meant to star Lancaster but he wound up not appearing in the film – the first of their productions in which he did not act.

Hecht-Lancaster Productions
In 1951, the actor/producer duo changed the company's name to Hecht-Lancaster Productions. The first film under the new name was another swashbuckler: 1952's The Crimson Pirate, directed by Siodmak. Co-starring Cravat, it was extremely popular.

Lancaster changed pace once more by doing a straight dramatic part in 1952's Come Back, Little Sheba, based on a Broadway hit, with Shirley Booth, produced by Wallis and directed by Daniel Mann.

Alternating with adventure films, he went into South Sea Woman in 1952 at Warners. Part of the Norma-Warners contract was that Lancaster had to appear in some non-Norma films, of which this was one.

In 1954, for his own company, Lancaster produced and starred in His Majesty O'Keefe, a South Sea island tale shot in Fiji. It was co-written by James Hill, who would soon become a part of the Hecht-Lancaster partnership.

United Artists
Hecht and Lancaster left Warners for United Artists, for what began as a two-picture deal, the first of which was to be 1954's Apache, starring Lancaster as a Native American.

They followed it with another Western in 1954, Vera Cruz, co-starring Gary Cooper and produced by Hill. Both films were directed by Robert Aldrich and were hugely popular.

United Artists signed Hecht-Lancaster to a multi-picture contract, to make seven films over two years. These included films in which Lancaster did not act. Their first was Marty in 1955, based on Paddy Chayefsky's TV play starring Ernest Borgnine and directed by Delbert Mann. It won both the Best Picture Oscar and the Palme d'Or award at Cannes and Borgnine an Best Actor Oscar. It also earned $2 million on a budget of $350,000. Vera Cruz had been a huge success, but Marty secured Hecht-Lancaster as one of the most successful independent production companies in Hollywood at the time. Marty star Borgnine was under contract to Hecht-Lancaster and was unhappy about his lack of upcoming roles, especially after only receiving some seven lines in 1957's Sweet Smell of Success and half of his normal pay for Marty. He eventually sued for breach of contract to gain back some of this money in 1957.

Without Hill, Hecht and Lancaster produced The Kentuckian in 1955. It was directed by Lancaster in his directorial debut, and he also played a lead role. Lancaster disliked directing and only did it once more, on 1974's The Midnight Man.

Lancaster still had commitments with Wallis, and made The Rose Tattoo for him in 1955, starring with Anna Magnani and Daniel Mann directing. It was very popular at the box office and critically acclaimed, winning Magnani an Oscar.

Hecht-Hill-Lancaster
In 1955, Hill was made an equal partner in Hecht-Lancaster, with his name added to the production company. Hecht-Hill-Lancaster (HHL) released their first film Trapeze in 1956, with Lancaster performing many of his own stunts. The film, co-starring Tony Curtis and Gina Lollobrigida, went on to become the production company's top box office success, and United Artists expanded its deal with HHL.

In 1956, Lancaster and Hecht partnered with Loring Buzzell and entered the music industry with the music publishing companies Leigh Music, Hecht-Lancaster & Buzzell Music, Calyork Music and Colby Music and the record labels Calyork Records and Maine Records.

The HHL team impressed Hollywood with its success; as Life wrote in 1957, "[a]fter the independent production of a baker's dozen of pictures, it has yet to have its first flop ... (They were also good pictures.)." In late 1957, they announced they would make ten films worth $14 million in 1958.

Lancaster made two films for Wallis to complete his eight-film commitment for that contract: The Rainmaker (1956) with Katharine Hepburn, which earned Lancaster a Golden Globe nomination for Best Actor; and Gunfight at the O.K. Corral (1957) with Kirk Douglas, which was a huge commercial hit directed by John Sturges.

Lancaster re-teamed with Tony Curtis in 1957 for Sweet Smell of Success, a co-production between Hecht-Hill-Lancaster and Curtis' own company with wife Janet Leigh, Curtleigh Productions. The movie, directed by Alexander Mackendrick, was a critical success but a commercial disappointment. Over the years it has come to be regarded as one of Lancaster's greatest films.

HHL produced seven additional films in the late 1950s. Four starred Lancaster: Run Silent, Run Deep (1958), a Robert Wise directed war film with Clark Gable, which was mildly popular; Separate Tables (1958) a hotel-set drama with Kerr and Rita Hayworth (who married James Hill), which received an Oscar nomination for Best Picture and Oscar awards for lead actor David Niven and supporting actress Wendy Hiller, and was both a critical and commercial success; The Devil's Disciple (1959), with Douglas and Laurence Olivier, which lost money (and saw Lancaster fire Mackendrick during shooting); and the Western The Unforgiven (1960), with Audrey Hepburn, which was a critical and commercial disappointment.

Three were made without Lancaster, all of which lost money: The Bachelor Party (1957), from another TV play by Chayefsky, and directed by Delbert Mann; Take a Giant Step (1959), about a black student; and Summer of the Seventeenth Doll (1960), from an Australian play, shot on location in Australia and Britain.

HHL served as the production company for the 1960–61 TV series Whiplash.

The Hecht-Hill-Lancaster Productions company dissolved in 1960 after Hill ruptured his relationship with both Hecht and Lancaster. Hill went on to produce a single additional film, The Happy Thieves, in a new production company, Hillworth Productions, co-owned with his wife Rita Hayworth.

Hecht and Lancaster
Lancaster played the title role in Elmer Gantry (1960), written and directed by Richard Brooks for United Artists. The film received five Academy Award nominations, including Best Picture and Best Actor. Lancaster won the 1960 Academy Award for Best Actor, a Golden Globe Award, and the New York Film Critics Award for his performance.

Hecht and Lancaster worked together on The Young Savages (1961), directed by John Frankenheimer and produced by Hecht. Sydney Pollack worked as a dialogue coach.

Lancaster starred in Judgment at Nuremberg (1961) for Stanley Kramer, alongside Spencer Tracy, Richard Widmark and a number of other stars. The film was both a commercial and critical success, receiving eleven Oscar nominations, including Best Picture.

He then did another film with Hecht and Frankenheimer (replacing Charles Crichton), Birdman of Alcatraz (1962), a largely fictionalized biography. In it he plays Robert Stroud, a federal prisoner incarcerated for life for two murders, who begins to collect birds and over time becomes an expert in bird diseases, even publishing a book. The film shows Stroud transferred to the maximum security Alcatraz prison where he is not allowed to keep birds and as he ages he gets married, markets bird remedies, helps stop a prison rebellion, and writes a book on the history of the U.S. penal system, but never gets paroled.

The sympathetic performance earned Lancaster a Best Actor Oscar nomination, a BAFTA Award for Best Actor, and a Golden Globe nomination for Best Actor in a Dramatic Role. Hecht went on to produce five films without Lancaster's assistance, through his company Harold Hecht Films Productions between 1961 and 1967, including another Academy Award winner, Cat Ballou, starring Lee Marvin and Jane Fonda.

Collaborations with younger filmmakers
Lancaster made A Child Is Waiting (1963) with Judy Garland. It was produced by Kramer and directed by John Cassavetes.

He went to Italy to star in The Leopard (1963) for Luchino Visconti, co-starring Alain Delon and Claudia Cardinale. It was one of Lancaster's favourite films and was a big hit in France but failed in the US (though the version released was much truncated).

He had a small role in The List of Adrian Messenger (1963) for producer/star Kirk Douglas, and then did two for Frankenheimer: Seven Days in May (1964), a political thriller with Douglas, and The Train (1964), a World War Two action film (Lancaster had Frankenheimer replace Arthur Penn several days into filming).

Lancaster starred in The Hallelujah Trail (1965), a comic Western produced and directed by John Sturges which failed to recoup its large cost.

He had a big hit with The Professionals (1966), a Western directed by Brooks and also starring Lee Marvin.

In 1966, at the age of 52, Lancaster appeared nude in director Frank Perry's film The Swimmer (1968), in what the critic Roger Ebert called "his finest performance". Prior to working on The Swimmer, Lancaster was terrified of the water because he did not know how to swim. In preparation for the film, he took swimming lessons from UCLA swim coach Bob Horn. Filming was difficult and clashes between Lancaster and Perry led to Sydney Pollack coming in to do some filming. The film was not released until 1968, when it proved to be a commercial failure, though Lancaster remained proud of the movie and his performance.

Norlan Productions
In 1967, Lancaster formed a new partnership with Roland Kibbee, who had already worked as a writer on five Lancaster projects: Ten Tall Men, The Crimson Pirate, Three Sailors and a Girl (in which Lancaster made a cameo appearance), Vera Cruz, and The Devil's Disciple.

Through Norlan Productions, Lancaster and Kibbee produced The Scalphunters in 1968, directed by Sydney Pollack.

Lancaster followed it with another film from Pollack, Castle Keep in 1969, which was a big flop. So was The Gypsy Moths, for Frankenheimer, also in 1969.

1970s
Lancaster had one of the biggest successes of his career with Airport in 1970, starring alongside Dean Martin, George Kennedy, Van Heflin, Helen Hayes, Maureen Stapleton, Barbara Hale, Jean Seberg, and Jacqueline Bisset. The Ross Hunter film received nine Academy Award nominations, including one for Best Picture. It became one of the biggest box-office hits of 1970 and, at that time, reportedly the highest-grossing film in the history of Universal Pictures.

He then went into a series of Westerns: Lawman in 1971, directed by Michael Winner; Valdez Is Coming in 1971, for Norlan; and Ulzana's Raid in 1972, directed by Aldrich and produced by himself and Hecht. None were particularly popular but Ulzana's Raid has become a cult film.

Lancaster did two thrillers, both 1973: Scorpio with Winner and Executive Action.

Lancaster returned to directing in 1974 with The Midnight Man, which he also wrote and produced with Kibee.

He made a second film with Visconti, Conversation Piece in 1974 and played the title role in the TV series Moses the Lawgiver, also in 1974.

Lancaster was one of many names in 1975's 1900, directed by Bernardo Bertolucci, and he had a cameo in 1976's Buffalo Bill and the Indians, or Sitting Bull's History Lesson for Robert Altman.

He played Shimon Peres in the TV movie Victory at Entebbe in 1977 and had a supporting role in The Cassandra Crossing in 1976. He made a fourth and final film with Aldrich, Twilight's Last Gleaming in 1977, and had the title role in 1977's The Island of Dr. Moreau.

Lancaster was top-billed in Go Tell the Spartans in 1978, a Vietnam War film; Lancaster admired the script so much that he took a reduced fee and donated money to help the movie to be completed. He was in Zulu Dawn in 1979.

1980s
Lancaster began the 1980s with a highly acclaimed performance alongside Susan Sarandon in Atlantic City in 1980, directed by Louis Malle. The film received five Oscar nominations, including Best Picture and a Best Actor nomination for Lancaster.

He had key roles in Cattle Annie and Little Britches in 1981, The Skin in 1982 with Cardinale, Marco Polo, also in 1982, and Local Hero in 1983.

By now, Lancaster was mostly a character actor in features, as in The Osterman Weekend in 1983, but he was the lead in the TV movie Scandal Sheet in 1985.

He was in Little Treasure in 1985, directed by Alan Sharp, who had written Ulzana's Raid; On Wings of Eagles for TV in 1986, as Bull Simons; 1986's made for TV Barnum starred him in the title role; Tough Guys reunited him on the big screen with Kirk Douglas in 1986; Fathers and Sons: A German Tragedy (in German Väter und Söhne – Eine deutsche Tragödie) in 1986 for German TV; 1987's Control made in Italy; Rocket Gibraltar in 1988, and The Jeweller's Shop in 1989.

His first critical success in a while was Field of Dreams in 1989, in which he played a supporting role as Moonlight Graham. He was also in the miniseries The Betrothed in 1989.

Later career
Lancaster's final performances included TV miniseries The Phantom of the Opera (1990); Voyage of Terror: The Achille Lauro Affair (1990) as Leon Klinghoffer based on the 1985 MS Achille Lauro hijacking incident; and Separate But Equal (1991) with Sidney Poitier.

Frequent collaborators
Lancaster appeared in a total of seventeen films produced by his agent, Harold Hecht. Eight of these were co-produced by James Hill. He also appeared in eight films produced by Hal B. Wallis and two with producer Mark Hellinger. Although Lancaster's work alongside Kirk Douglas was known as that of a successful pair of actors, Douglas, in fact, produced four films for the pair, through his production companies Bryna Productions and Joel Productions. Roland Kibbee also produced three Lancaster films, and Lancaster was also cast in two Stanley Kramer productions.

Kirk Douglas
Kirk Douglas starred in seven films across the decades with Burt Lancaster: I Walk Alone (1948), Gunfight at the O.K. Corral (1957), The Devil's Disciple (1959), The List of Adrian Messenger (1963), Seven Days in May (1964), Victory at Entebbe (1976) and Tough Guys (1986), which fixed the notion of the pair as something of a team in the public imagination. Douglas was always billed under Lancaster in these movies but, with the exception of I Walk Alone, in which Douglas played a villain, their roles were usually more or less the same size. Both actors arrived in Hollywood at about the same time, and first appeared together in the fourth film for each, albeit with Douglas in a supporting role. They both became actor-producers who sought out independent Hollywood careers.

John Frankenheimer
John Frankenheimer directed five films with Lancaster: The Young Savages (1961), Birdman of Alcatraz (1962), Seven Days in May (1964), The Train (1964), and The Gypsy Moths (1969).

Other repeat collaborators
He was directed four times by Robert Aldrich, three times each by Robert Siodmak and Sydney Pollack, and twice each by Byron Haskin, Daniel Mann, John Sturges, John Huston, Richard Brooks, Alexander Mackendrick, Luchino Visconti, and Michael Winner.

Roland Kibbee wrote for seven Lancaster films. Lancaster used makeup veteran Robert Schiffer in twenty credited films, hiring Schiffer on nearly all of the films he produced.

Political activism
Lancaster was a vocal supporter of progressive and liberal political causes. He frequently spoke out in support of racial and other minorities. As a result, he was often a target of FBI investigations. He was named in President Richard Nixon's 1973 "Enemies List".

A vocal opponent of the Vietnam War, he helped pay for the successful defense of a soldier accused of "fragging" (i.e., murdering) another soldier during war-time. In 1968, Lancaster actively supported the presidential candidacy of anti-war Senator Eugene McCarthy of Minnesota, and frequently spoke on his behalf during the Democratic primaries.

Lancaster was also active in anti-death penalty activism. He campaigned heavily for George McGovern in the 1972 United States presidential election.

In 1985, Lancaster joined the fight against AIDS after fellow movie star Rock Hudson contracted the disease. Lancaster delivered Hudson's last words at the Commitment to Life fundraiser at a time when the stigma surrounding AIDS was at its height.

Of his political opinions, frequent co-star Tony Curtis said: "Here's this great big aggressive guy that looks like a ding-dong athlete playing these big tough guys and he has the soul of—who were those first philosophers of equality?—Socrates, Plato. He was a Greek philosopher with a sense that everybody was equal."

Actor and SAG president Ed Asner said he showed everybody in Hollywood "how to be a liberal with balls".

Hollywood Ten
In 1947, Lancaster reportedly signed a statement release by the National Council of Arts, Sciences and Professions (NCASP) asking Congress to abolish the House Un-American Activities Committee (HUAC). He was also a member of the short-lived Committee for the First Amendment, formed in support of the Hollywood Ten. He was one of 26 movie stars who flew to Washington in October 1947 to protest against the HUAC hearings. The committee's Hollywood Fights Back broadcasts on ABC Radio Network were two 30-minute programs that took place on October 27 and November 2, 1947, during which committee members voiced their opposition to the HUAC hearings. Many members faced blacklisting and backlash due to their involvement in the committee. Lancaster was listed in anti-communist literature as a fellow traveler.

Civil rights movement
He and his second wife, Norma, hosted a fundraiser for Martin Luther King Jr. and the Student Diversity Leadership Conference (SDLC) ahead of the historic March on Washington in 1963. He attended the march, where he was one of the speakers. He flew in from France for the event, where he was shooting The Train, and flew back again the next day, despite a reported fear of flying.

On August 28, 1963, at the March on Washington Lancaster "read the speech that James Baldwin was supposed to make," because (as Malcolm X said in a speech delivered in Detroit at the King Solomon Baptist Church in late 1963) "they wouldn't let Baldwin get up there because they know Baldwin is liable to say anything."

ACLU
In 1968, Lancaster was elected to serve as chairman of the Roger Baldwin Foundation, a newly formed fund-raising arm of the American Civil Liberties Union of Southern California. His co-chairs were Frank Sinatra and Irving L. Lichtenstein. In October 1968, he hosted a party at his home to raise money for the ACLU to use for the defense of the more than four hundred people arrested at the 1968 Democratic National Convention. Throughout the years, he remained an ardent supporter and a fundraiser for the organization.

While serving as a member of the five-person ACLU Foundation executive committee, he cast the key vote to retain Ramona Ripston as executive director of the Southern California affiliate, a position she would build into a powerful advocacy force in Los Angeles politics. Ripston later recalled: "There was a feeling that a woman couldn't run the ACLU foundation, nor have access to the books. The vote finally came down to two 'yes' and two 'no.' Who had the deciding vote? Burt. He had a scotch or two and finally he said, 'I think she should be executive director.' I always loved him for that."

When President George H. W. Bush derided Democratic candidate Michael Dukakis as a "card-carrying member of the ACLU", Lancaster was one of the supporters featured in the organization's first television advertising campaign stating: "I'm a card-carrying member of the ACLU" and "No one agrees with every single thing they've done. But no one can disagree with the guiding principle—with liberty and justice for all.'" He also campaigned for Michael Dukakis in the 1988 United States presidential election.

Marriages and relationships
Lancaster guarded his personal life and attempted to keep it private despite his stardom. He was married three times and had five children.

His first marriage was to June Ernst, a trapeze acrobat. Ernst was the daughter of a renowned female aerialist and an accomplished acrobat herself. After they were married, he performed with her family and her until their separation in the late 1930s. When they divorced is unclear. Contemporary reports listed 1940, but subsequent biographers have suggested dates as late as 1946, delaying his marriage to his second wife.

He met second wife Norma Anderson (1917–1988) when the stenographer substituted for an ill actress in a USO production for the troops in Italy. Reportedly, on seeing Lancaster in the crowd on her way to town from the airport, she turned to an officer and asked, "Who is that good-looking officer and is he married?" The officer set up a blind date between the two for that evening. They married in 1946. Norma was active in political causes with an entire room in their Bel Air home devoted to her major interest, the League of Woman Voters, crammed with printing presses and all the necessary supplies for mass mailings.

She was a life-long member of the NAACP. The couple held a fundraiser for Martin Luther King Jr. and the Southern Christian Leadership Conference ahead of the 1963 March on Washington. All five of his children were with Anderson: Bill (who became an actor and screenwriter), James, Susan, Joanna (who worked as a film producer), and Sighle (pronounced "Sheila"). It was a troubled marriage. The pair separated in 1966, and divorced in 1969.

In 1966, Lancaster began a long-term relationship with hairdresser Jackie Bone, who worked on The Professionals. The relationship was tempestuous, with Bone once smashing a wine bottle over Lancaster's head at a dinner with Sydney Pollack and Peter Falk. Reportedly, they eventually split up after her religious conversion, which Lancaster believed he could not share with her.

His third marriage, to Susan Martin, lasted from September 1990 until his death in 1994.

According to biographer Kate Buford in Burt Lancaster: An American Life, Lancaster was devotedly loyal to his friends and family. Old friends from his childhood remained his friends for life.

Possible affairs
Some media outlets and authors have written that Lancaster was bisexual and had relationships with both men and women.

Friends said he claimed he was romantically involved with Deborah Kerr during the filming of From Here to Eternity in 1953. However, Kerr stated that while there was a spark of attraction, nothing ever happened.

He reportedly had an affair with Joan Blondell.

In her 1980 autobiography, Shelley Winters claimed to have had a two-year affair with him, during which time he was considering separation from his wife. In his Hollywood memoirs, friend Farley Granger recalled an incident when Lancaster and he had to come to Winters' rescue one evening when she had inadvertently overdosed on alcohol and sleeping pills. She broke up with him for "cheating on her with his wife" after she heard reports of his wife's third or fourth pregnancy. Lancaster and Winters performed together in the 1949 radio play adaptation of The Killers. They appeared in two films together: The Young Savages, where she played his character's former lover, and The Scalphunters.

Religion
Despite his Protestant background and upbringing, Lancaster identified as an atheist later in life.

Health problems
As Lancaster reached his 60s, he began to be affected by cardiovascular disease. In January 1980, he had complications from a routine gall bladder operation (that he barely survived). In 1983, following two minor heart attacks, he underwent an emergency quadruple coronary bypass. He continued to act, however, and to engage in public activism. In 1988, he attended a congressional hearing in Washington, DC, with former colleagues who included James Stewart and Ginger Rogers to protest against media magnate Ted Turner's plan to colorize various black-and-white films from the 1930s and 1940s. On November 30, 1990, when he was 77, a stroke left him partially paralyzed and largely unable to speak, ending his acting career.

Death
Lancaster died at his apartment in Century City, Los Angeles, after having a third heart attack at 4:50 am on October 20, 1994. His body was cremated, and his ashes were scattered under a large oak tree in Westwood Memorial Park, which is located in Westwood Village, California. A small, square ground plaque amid several others, inscribed "Burt Lancaster 1913–1994", marks the location. As he had requested, no memorial or funeral service was held for him.

Legacy
The centennial of Lancaster's birth was honored at New York City's Film Society of Lincoln Center in May 2013 with the screening of 12 of the actor's best-known films, from The Killers to Atlantic City.

Lancaster has a star on the Hollywood Walk of Fame, at 6801 Hollywood Boulevard.

Filmography and awards
Lancaster was nominated for the Oscar for Best Actor in a Leading Role in 1954 for From Here to Eternity, in 1961 for Elmer Gantry, in 1964 for Birdman of Alcatraz, and in 1982 for Atlantic City and won the award in 1961. Lancaster's leading role in Luchino Visconti's 1963 canonical The Leopard began a series of roles with important European art film directors that included roles in Bernardo Bertolucci's 1900 and Louis Malle's Atlantic City as well as Visconti's Conversation Piece.

Box office ranking
For a number of years exhibitors voted Lancaster among the most popular stars:

In other media
Spanish music group Hombres G released an album named La cagaste, Burt Lancaster (You messed up, Burt Lancaster) in 1986.

Thomas Hart Benton painted a scene from The Kentuckian as part of the film's marketing. Lancaster posed for the painting, also known as The Kentuckian.

title: Burt Lancaster


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Talk:Blitz

Untitled
Blitz doesn't mean "very fast" in German. It simply means "flash" or "lightning". Probably the misunderstanding stems from the word "blitzschnell" meaning "fast as lightning".

The Blitz++ is under HEY, HOE YO DOING...the GNU General Public License and Artistic licence as it is written on the Legal page and can be freely copied. RedAndr 20:00, 11 May 2007 (UTC)
 * I disagree with your assessment. While the software is released under the GPL, there is no indication of a documentation release under GFDL. I don't think the content posted on sourceforge is reputed not having any copyright (talking about everything that is not a program, of course). -- lucasbfr talk 20:15, 11 May 2007 (UTC)
 * What about README file, which is distributed inside the program package? Can it be considered as a part of the program? May I cite it? RedAndr 21:48, 11 May 2007 (UTC)
 * Errrrr, good question. There might also be an issue on the notability of the program, by the way. Was it cited by independent reliable sources? If it was, don't forget to cite them in the article or it might be speedy deleted as not asserting the importance of the subject. -- lucasbfr talk 22:09, 11 May 2007 (UTC)
 * I gather from this page that the GPL is not compatible with the GFDL, so neither the readme nor the documentation on the website can be posted to Wikipedia without permission. --Butseriouslyfolks 23:56, 11 May 2007 (UTC)
 * From README: "Licensing information is at http://oonumerics.org/legal. Summary: you can do anything except sell this library in source form. Blitz is licensed under both the GPL (see COPYING) and a less restrictive (non-viral) "artistic license" (see LICENSE)." Concerning notability, it's not a problem, since the library is very widely used, there are a lot of independent sources. RedAndr 01:29, 12 May 2007 (UTC)
 * That is insufficient for use at Wikipedia. Neither the GPL nor the "artistic license" equal GFDL. --Butseriouslyfolks 06:57, 12 May 2007 (UTC)
 * I have received permission from Prof. Todd Veldhuizen, the creator of the library, to use texts from the Blitz++ site in Wikipedia. —The preceding unsigned comment was added by RedAndr (talk • contribs) 13:18, 15 May 2007 (UTC).

Back to the fist point: "blitz" means "lightning at the heaven within a thunderstorm" or "the flash at or beside a photocamera". And it does not mean "fast". There is one word - "blitzschnell" - meaning fast like a (thunderstorm-)flash. The word for "fast" in german is "schnell". And it is used like "fast" in englisch. Here is a example and the transcription word by word: "Das ist zu schnell" = "This is to fast"  or  "Das Auto ist schnell" = "The car is fast". It is impossible to use the word "blitz" as a word for "fast"! So I tried to edit the "intro". 95.157.17.217 (talk) 01:29, 18 February 2010 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Blitz


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Balts

The Balts or Baltic peoples (baltai, balti) are a group of peoples inhabiting the eastern coast of the Baltic Sea who speak Baltic languages. Among the Baltic peoples are modern-day Lithuanians (including Samogitians) and Latvians (including Latgalians) — all East Balts — as well as the Old Prussians, Curonians, Sudovians, Skalvians, Yotvingians and Galindians — the West Balts — whose languages and cultures are now extinct.

The Balts are descended from a group of Indo-European tribes who settled the area between the lower Vistula and southeast shore of the Baltic Sea and upper Daugava and Dnieper rivers, and which over time became differentiated into West and East Balts. In the fifth century CE parts of the eastern Baltic coast began to be settled by the ancestors of the Western Balts, whereas the East Balts lived in modern-day Belarus, Ukraine and Russia. In the first millennium CE, large migrations of the Balts occurred. By the 13th and 14th centuries, the East Balts shrank to the general area that the present-day Balts and Belarusians inhabit.

Baltic languages belong to the Balto-Slavic branch of the Indo-European languages. One of the features of Baltic languages is the number of conservative or archaic features retained.

Etymology
Medieval German chronicler Adam of Bremen in the latter part of the 11th century AD was the first writer to use the term "Baltic" in reference to the sea of that name. Before him various ancient places names, such as Balcia, were used in reference to a supposed island in the Baltic Sea.

Adam, a speaker of German, connected Balt- with belt, a word with which he was familiar.

In Germanic languages there was some form of the toponym East Sea until after about the year 1600, when maps in English began to label it as the Baltic Sea. By 1840, German nobles of the Governorate of Livonia adopted the term "Balts" to distinguish themselves from Germans of Germany. They spoke an exclusive dialect, Baltic German, which was regarded by many as the language of the Balts until 1919.

In 1845, Georg Heinrich Ferdinand Nesselmann proposed a distinct language group for Latvian, Lithuanian, and Old Prussian, which he termed Baltic. The term became prevalent after Latvia and Lithuania gained independence in 1918. Up until the early 20th century, either "Latvian" or "Lithuanian" could be used to mean the entire language family.

Origins
The Balts or Baltic peoples, defined as speakers of one of the Baltic languages, a branch of the Indo-European language family, are descended from a group of Indo-European tribes who settled the area between the lower Vistula and southeast shore of the Baltic Sea and upper Daugava and Dnieper rivers. The Baltic languages, especially Lithuanian, retain a number of conservative or archaic features, perhaps because the areas in which they are spoken are geographically consolidated and have low rates of immigration.

Some of the major authorities on Balts, such as Kazimieras Būga, Max Vasmer, Vladimir Toporov and Oleg Trubachyov, in conducting etymological studies of eastern European river names, were able to identify in certain regions names of specifically Baltic provenance, which most likely indicate where the Balts lived in prehistoric times. According to Vladimir Toporov and Oleg Trubachyov, the eastern boundary of the Balts in the prehistoric times were the upper reaches of the Volga, Moskva, and Oka rivers, while the southern border was the Seym river. This information is summarized and synthesized by Marija Gimbutas in The Balts (1963) to obtain a likely proto-Baltic homeland. Its borders are approximately: from a line on the Pomeranian coast eastward to include or nearly include the present-day sites of Berlin, Warsaw, Kyiv, and Kursk, northward through Moscow to the River Berzha, westward in an irregular line to the coast of the Gulf of Riga, north of Riga.

However, other scholars such as Endre Bojt (1999) reject the presumption that there ever was such a thing as a clear, single "Baltic Urheimat": "'The references to the Balts at various Urheimat locations across the centuries are often of doubtful authenticity, those concerning the Balts furthest to the West are the more trustworthy among them. (...) It is wise to group the particulars of Baltic history according to the interests that moved the pens of the authors of our sources.'"

Proto-history
The area of Baltic habitation shrank due to assimilation by other groups, and invasions. According to one of the theories which has gained considerable traction over the years, one of the western Baltic tribes, the Galindians, Galindae, or Goliad, migrated to the area around modern-day Moscow, Russia around the fourth century AD.

Over time the Balts became differentiated into West and East Balts. In the fifth century AD parts of the eastern Baltic coast began to be settled by the ancestors of the Western Balts: Brus/Prūsa ("Old Prussians"), Sudovians/Jotvingians, Scalvians, Nadruvians, and Curonians. The East Balts, including the hypothesised Dniepr Balts, were living in modern-day Belarus, Ukraine and Russia.

Germanic peoples lived to the west of the Baltic homelands; by the first century AD, the Goths had stabilized their kingdom from the mouth of the Vistula, south to Dacia. As Roman domination collapsed in the first half of the first millennium CE in Northern and Eastern Europe, large migrations of the Balts occurred — first, the Galindae or Galindians towards the east, and later, East Balts towards the west. In the eighth century, Slavic tribes from the Volga regions appeared. By the 13th and 14th centuries, they reached the general area that the present-day Balts and Belarusians inhabit. Many other Eastern and Southern Balts either assimilated with other Balts, or Slavs in the fourth–seventh centuries and were gradually slavicized.

Middle Ages
In the 12th and 13th centuries, internal struggles and invasions by Ruthenians and Poles, and later the expansion of the Teutonic Order, resulted in an almost complete annihilation of the Galindians, Curonians, and Yotvingians. Gradually, Old Prussians became Germanized or Lithuanized between the 15th and 17th centuries, especially after the Reformation in Prussia. The cultures of the Lithuanians and Latgalians/Latvians survived and became the ancestors of the populations of the modern-day countries of Latvia and Lithuania.

Old Prussian was closely related to the other extinct Western Baltic languages, Curonian, Galindian and Sudovian. It is more distantly related to the surviving Eastern Baltic languages, Lithuanian and Latvian. Compare the Prussian word seme (zemē), Latvian zeme, the Lithuanian žemė (land in English).

Culture
The Balts originally practiced Baltic religion. They were gradually Christianized as a result of the Northern Crusades of the Middle Ages. Baltic peoples such as the Latvians, Lithuanians and Old Prussians had their distinct mythologies. The Lithuanians have close historic ties to Poland, and many of them are Roman Catholic. The Latvians have close historic ties to Northern Germany and Scandinavia, and many of them are irreligious. In recent times, the Baltic religion has been revived in Baltic neopaganism.

Genetics
The Balts are included in the "North European" gene cluster together with the Germanic peoples, some Slavic groups (the Poles and Northern Russians) and Baltic Finnic peoples.

Saag et a. (2017) detected that the eastern Baltic in the Mesolithic was inhabited primarily by Western Hunter-Gatherers (WHGs). Their paternal haplogroups were mostly types of I2a and R1b, while their maternal haplogroups were mostly types of U5, U4 and U2. These people carried a high frequency of the derived HERC2 allele which codes for light eye color and possess an increased frequency of the derived alleles for SLC45A2 and SLC24A5, coding for lighter skin color.

Baltic hunter-gatherers still displayed a slightly larger amount of WHG ancestry than Scandinavian Hunter-Gatherers (SHGs). WHG ancestry in the Baltic was particularly high among hunter-gatherers in Latvia and Lithuania. Unlike other parts of Europe, the hunter-gatherers of the eastern Baltic do not appear to have mixed much with Early European Farmers (EEFs) arriving from Anatolia.

During the Neolithic, increasing admixture from Eastern Hunter-Gatherers (EHGs) is detected. The paternal haplogroups of EHGs was mostly types of R1a, while their maternal haplogroups appears to have been almost exclusively types of U5, U4, and U2.

The rise of the Corded Ware culture in the eastern Baltic in the Chalcolithic and Bronze Age is accompanied by a significant infusion of steppe ancestry and EEF ancestry into the eastern Baltic gene pool. In the aftermath of the Corded Ware expansion, local hunter-gatherer ancestry experienced a resurgence.

Haplogroup N reached the eastern Baltic only in the Late Bronze Age, probably with the speakers of the Uralic languages.

Modern-day Balts have a lower amount of EEF ancestry, and a higher amount of WHG ancestry, than any other population in Europe.

List of Baltic peoples
Modern-day Baltic peoples
 * East Baltic peoples
 * Latvians
 * Latgalians
 * Lithuanians
 * Aukštaitija ("highlanders")
 * Samogitians ("lowlanders")

title: Balts


text:
Burnt-in timecode

Burnt-in timecode (often abbreviated to BITC by analogy to VITC) is a human-readable on-screen version of the timecode information for a piece of material superimposed on a video image. BITC is sometimes used in conjunction with "real" machine-readable timecode but more often used in copies of original material onto a nonbroadcast format such as VHS so that the VHS copies can be traced back to their master tape and the original timecodes easily located.

Many professional VTRs can "burn" (overlay) the tape timecode onto one of their outputs. This output (which usually also displays the setup menu or on-screen display) is known as the super out or monitor out. The character switch or menu item turns this behaviour on or off. The character function also displays the timecode on the preview monitors in linear editing suites.

Videotapes that are recorded with timecode numbers overlaid on the video are referred to as window dubs, named after the "window" that displays the burnt-in timecode on-screen.

When editing was done using magnetic tapes that were subject to damage from excessive wear, it was common to use a window dub as a working copy for the majority of the editing process. Editing decisions would be made using a window dub, and no specialized equipment was needed to write down an edit decision list, which would then be replicated from the high-quality masters.

Timecode can also be superimposed on video using a dedicated overlay device, often called a "window dub inserter". This inputs a video signal and its separate timecode audio signal, reads the timecode, superimposes the timecode display over the video, and outputs the combined display (usually via composite), all in real time. Stand-alone timecode generators/readers often have the window dub function built in.

Some consumer cameras, in particular DV cameras, can "burn" (overlay) the tape timecode onto the composite output. This output is typically semitransparent and may include other tape information. It is usually activated by turning on the "display" info in one of the camera's submenus. While not as "professional" as an overlay created by a professional VCR, it is a cheap alternative that is just as accurate.

Timecode is stored in the metadata areas of captured DV AVI files, and some software is able to "burn" (overlay) this into the video frames. For example, DVMP Pro is able to "burn" timecode or other items of DV metadata (such as date and time, iris, shutter speed, gain, white balance mode, etc.) into DV AVI files.

OCR techniques can be used to read BITC in situations where other forms of timecode are not available.

title: Burnt-in timecode


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Bra–ket notation

Bra–ket notation, also called Dirac notation, is a notation for linear algebra and linear operators on complex vector spaces together with their dual space both in the finite-dimensional and infinite-dimensional case. It is specifically designed to ease the types of calculations that frequently come up in quantum mechanics. Its use in quantum mechanics is quite widespread.

Bra–ket notation was created by Paul Dirac in his 1939 publication A New Notation for Quantum Mechanics. The notation was introduced as an easier way to write quantum mechanical expressions. The name comes from the English word "bracket".

Quantum mechanics
In quantum mechanics, bra–ket notation is used ubiquitously to denote quantum states. The notation uses angle brackets, $\langle$ and $\rangle$, and a vertical bar $, to construct "bras" and "kets".

A ket is of the form $$|v \rangle$$. Mathematically it denotes a vector, $$\boldsymbol v$$, in an abstract (complex) vector space $$V$$, and physically it represents a state of some quantum system.

A bra is of the form $$\langle f|$$. Mathematically it denotes a linear form $$f:V \to \Complex$$, i.e. a linear map that maps each vector in $$V$$ to a number in the complex plane $$\Complex$$. Letting the linear functional $$ \langle f|$$ act on a vector $$|v\rangle$$ is written as $$\langle f | v\rangle \in \Complex$$.

Assume that on $$V$$ there exists an inner product $$(\cdot,\cdot)$$ with antilinear first argument, which makes $$V$$ an inner product space. Then with this inner product each vector $$\boldsymbol \phi \equiv |\phi\rangle$$ can be identified with a corresponding linear form, by placing the vector in the anti-linear first slot of the inner product: $$(\boldsymbol\phi,\cdot) \equiv \langle\phi|$$. The correspondence between these notations is then $$(\boldsymbol\phi, \boldsymbol\psi) \equiv \langle\phi|\psi\rangle$$. The linear form $$\langle\phi|$$ is a covector to $$|\phi\rangle$$, and the set of all covectors forms a subspace of the dual vector space $$ V^\vee$$, to the initial vector space $$V$$. The purpose of this linear form $$\langle\phi|$$ can now be understood in terms of making projections onto the state $$\boldsymbol \phi,$$ to find how linearly dependent two states are, etc.

For the vector space $$\Complex^n$$, kets can be identified with column vectors, and bras with row vectors. Combinations of bras, kets, and linear operators are interpreted using matrix multiplication. If $$\Complex^n$$ has the standard Hermitian inner product $$(\boldsymbol v, \boldsymbol w) = v^\dagger w$$, under this identification, the identification of kets and bras and vice versa provided by the inner product is taking the Hermitian conjugate (denoted $$ \dagger$$).

It is common to suppress the vector or linear form from the bra–ket notation and only use a label inside the typography for the bra or ket. For example, the spin operator $$\hat \sigma_z$$ on a two-dimensional space $$\Delta$$ of spinors has eigenvalues $\pm \frac{1}{2}$ with eigenspinors $$\boldsymbol \psi_+,\boldsymbol \psi_- \in \Delta$$. In bra–ket notation, this is typically denoted as $$\boldsymbol \psi_+ = |+\rangle$$, and $$\boldsymbol \psi_- = |-\rangle$$. As above, kets and bras with the same label are interpreted as kets and bras corresponding to each other using the inner product. In particular, when also identified with row and column vectors, kets and bras with the same label are identified with Hermitian conjugate column and row vectors.

Bra–ket notation was effectively established in 1939 by Paul Dirac; it is thus also known as Dirac notation, despite the notation having a precursor in Hermann Grassmann's use of $$[\phi{\mid}\psi]$$ for inner products nearly 100 years earlier.

Vectors vs kets
In mathematics, the term "vector" is used for an element of any vector space. In physics, however, the term "vector" tends to refer almost exclusively to quantities like displacement or velocity, which have components that relate directly to the three dimensions of space, or relativistically, to the four of spacetime. Such vectors are typically denoted with over arrows ($$\vec r$$), boldface ($$\mathbf{p}$$) or indices ($$v^\mu$$).

In quantum mechanics, a quantum state is typically represented as an element of a complex Hilbert space, for example, the infinite-dimensional vector space of all possible wavefunctions (square integrable functions mapping each point of 3D space to a complex number) or some more abstract Hilbert space constructed more algebraically. To distinguish this type of vector from those described above, it is common and useful in physics to denote an element $$\phi$$ of an abstract complex vector space as a ket $$|\phi\rangle$$, to refer to it as a "ket" rather than as a vector, and to pronounce it "ket-$$\phi$$" or "ket-A" for $|A\rangle$.

Symbols, letters, numbers, or even words—whatever serves as a convenient label—can be used as the label inside a ket, with the $$|\ \rangle$$ making clear that the label indicates a vector in vector space. In other words, the symbol "$|A\rangle$" has a recognizable mathematical meaning as to the kind of variable being represented, while just the "$A$" by itself does not. For example, $|1\rangle + |2\rangle$ is not necessarily equal to $|3\rangle$. Nevertheless, for convenience, there is usually some logical scheme behind the labels inside kets, such as the common practice of labeling energy eigenkets in quantum mechanics through a listing of their quantum numbers. At its simplest, the label inside the ket is the eigenvalue of a physical operator, such as $$\hat x$$, $$\hat p$$, $$\hat L_z$$, etc.

Notation
Since kets are just vectors in a Hermitian vector space, they can be manipulated using the usual rules of linear algebra. For example:

 * $$\begin{align}

\end{align}$$
 * A \rangle &= |B\rangle + |C\rangle \\
 * C \rangle &= (-1+2i)|D \rangle \\
 * D \rangle &= \int_{-\infty}^{\infty} e^{-x^2} |x\rangle \, \mathrm{d}x \,.

Note how the last line above involves infinitely many different kets, one for each real number $x$.

Since the ket is an element of a vector space, a bra $$\langle A|$$ is an element of its dual space, i.e. a bra is a linear functional which is a linear map from the vector space to the complex numbers. Thus, it is useful to think of kets and bras as being elements of different vector spaces (see below however) with both being different useful concepts.

A bra $$\langle\phi|$$ and a ket $$ |\psi\rangle$$ (i.e. a functional and a vector), can be combined to an operator $$|\psi\rangle\langle\phi|$$ of rank one with outer product

 * $$|\psi\rangle\langle\phi| \colon |\xi\rangle \mapsto |\psi\rangle\langle\phi|\xi\rangle ~.$$

Inner product and bra–ket identification on Hilbert space
The bra–ket notation is particularly useful in Hilbert spaces which have an inner product that allows Hermitian conjugation and identifying a vector with a continuous linear functional, i.e. a ket with a bra, and vice versa (see Riesz representation theorem). The inner product on Hilbert space $$(\, \ )$$ (with the first argument anti linear as preferred by physicists) is fully equivalent to an (anti-linear) identification between the space of kets and that of bras in the bra ket notation: for a vector ket $$\phi = |\phi\rangle $$ define a functional (i.e. bra) $$f_\phi = \langle\phi|$$ by

 * $$(\phi,\psi) = (|\phi\rangle, |\psi\rangle) =: f_\phi(\psi) = \langle\phi| \, \bigl(|\psi\rangle\bigr) =: \langle\phi{\mid}\psi\rangle $$

Bras and kets as row and column vectors
In the simple case where we consider the vector space $$\Complex^n$$, a ket can be identified with a column vector, and a bra as a row vector. If moreover we use the standard Hermitian inner product on $$\Complex^n$$, the bra corresponding to a ket, in particular a bra $\langlem|$ and a ket $|m\rangle$ with the same label are conjugate transpose. Moreover, conventions are set up in such a way that writing bras, kets, and linear operators next to each other simply imply matrix multiplication. In particular the outer product $$|\psi\rangle\langle\phi| $$ of a column and a row vector ket and bra can be identified with matrix multiplication (column vector times row vector equals matrix).

For a finite-dimensional vector space, using a fixed orthonormal basis, the inner product can be written as a matrix multiplication of a row vector with a column vector: $$ \langle A | B \rangle \doteq A_1^* B_1 + A_2^* B_2 + \cdots + A_N^* B_N = \begin{pmatrix} A_1^* & A_2^* & \cdots & A_N^* \end{pmatrix} \begin{pmatrix} B_1 \\ B_2 \\ \vdots \\ B_N \end{pmatrix}$$ Based on this, the bras and kets can be defined as: $$\begin{align} \langle A | &\doteq \begin{pmatrix} A_1^* & A_2^* & \cdots & A_N^* \end{pmatrix} \\ \end{align}$$ and then it is understood that a bra next to a ket implies matrix multiplication.
 * B \rangle &\doteq \begin{pmatrix} B_1 \\ B_2 \\ \vdots \\ B_N \end{pmatrix}

The conjugate transpose (also called Hermitian conjugate) of a bra is the corresponding ket and vice versa: $$\langle A |^\dagger = |A \rangle, \quad |A \rangle^\dagger = \langle A |$$ because if one starts with the bra $$\begin{pmatrix} A_1^* & A_2^* & \cdots & A_N^* \end{pmatrix} \,,$$ then performs a complex conjugation, and then a matrix transpose, one ends up with the ket $$\begin{pmatrix} A_1 \\ A_2 \\ \vdots \\ A_N \end{pmatrix}$$

Writing elements of a finite dimensional (or mutatis mutandis, countably infinite) vector space as a column vector of numbers requires picking a basis. Picking a basis is not always helpful because quantum mechanics calculations involve frequently switching between different bases (e.g. position basis, momentum basis, energy eigenbasis), and one can write something like "$|m\rangle$" without committing to any particular basis. In situations involving two different important basis vectors, the basis vectors can be taken in the notation explicitly and here will be referred simply as "$|−\rangle$" and "$|+\rangle$".

Non-normalizable states and non-Hilbert spaces
Bra–ket notation can be used even if the vector space is not a Hilbert space.

In quantum mechanics, it is common practice to write down kets which have infinite norm, i.e. non-normalizable wavefunctions. Examples include states whose wavefunctions are Dirac delta functions or infinite plane waves. These do not, technically, belong to the Hilbert space itself. However, the definition of "Hilbert space" can be broadened to accommodate these states (see the Gelfand–Naimark–Segal construction or rigged Hilbert spaces). The bra–ket notation continues to work in an analogous way in this broader context.

Banach spaces are a different generalization of Hilbert spaces. In a Banach space $\mathcal{B}$, the vectors may be notated by kets and the continuous linear functionals by bras. Over any vector space without topology, we may also notate the vectors by kets and the linear functionals by bras. In these more general contexts, the bracket does not have the meaning of an inner product, because the Riesz representation theorem does not apply.

Usage in quantum mechanics
The mathematical structure of quantum mechanics is based in large part on linear algebra:
 * Wave functions and other quantum states can be represented as vectors in a complex Hilbert space. (The exact structure of this Hilbert space depends on the situation.) In bra–ket notation, for example, an electron might be in the "state" $|ψ\rangle$. (Technically, the quantum states are rays of vectors in the Hilbert space, as $c|ψ\rangle$ corresponds to the same state for any nonzero complex number $c$.)
 * Quantum superpositions can be described as vector sums of the constituent states. For example, an electron in the state $1⁄√2|1\rangle + i⁄√2|2\rangle$ is in a quantum superposition of the states $|1\rangle$ and $|2\rangle$.
 * Measurements are associated with linear operators (called observables) on the Hilbert space of quantum states.
 * Dynamics are also described by linear operators on the Hilbert space. For example, in the Schrödinger picture, there is a linear time evolution operator $U$ with the property that if an electron is in state $|ψ\rangle$ right now, at a later time it will be in the state $U|ψ\rangle$, the same $U$ for every possible $|ψ\rangle$.
 * Wave function normalization is scaling a wave function so that its norm is 1.

Since virtually every calculation in quantum mechanics involves vectors and linear operators, it can involve, and often does involve, bra–ket notation. A few examples follow:

Spinless position–space wave function
The Hilbert space of a spin-0 point particle is spanned by a "position basis" $k$, where the label $x$ extends over the set of all points in position space. This label is the eigenvalue of the position operator acting on such a basis state, $$ \hat{\mathbf{r}}|\mathbf{r}\rangle = \mathbf{r}|\mathbf{r}\rangle $$. Since there are an uncountably infinite number of vector components in the basis, this is an uncountably infinite-dimensional Hilbert space. The dimensions of the Hilbert space (usually infinite) and position space (usually 1, 2 or 3) are not to be conflated.

Starting from any ket $A_{k}$ in this Hilbert space, one may define a complex scalar function of $|A\rangle = Σ_{k} A_{k} |e_{k}\rangle$, known as a wavefunction, $$\Psi(\mathbf{r}) \ \stackrel{\text{def}}{=}\ \lang \mathbf{r}|\Psi\rang \,.$$

On the left-hand side, $k$ is a function mapping any point in space to a complex number; on the right-hand side, $$\left|\Psi\right\rangle = \int d^3\mathbf{r} \, \Psi(\mathbf{r}) \left|\mathbf{r}\right\rangle$$ is a ket consisting of a superposition of kets with relative coefficients specified by that function.

It is then customary to define linear operators acting on wavefunctions in terms of linear operators acting on kets, by $$\hat A(\mathbf{r}) ~ \Psi(\mathbf{r}) \ \stackrel{\text{def}}{=}\ \lang \mathbf{r}|\hat A|\Psi\rang \,.$$

For instance, the momentum operator $$\hat \mathbf {p}$$ has the following coordinate representation, $$\hat{\mathbf{p} } (\mathbf{r}) ~ \Psi(\mathbf{r}) \ \stackrel{\text{def}}{=}\ \lang \mathbf{r} |\hat \mathbf{p}|\Psi\rang = - i \hbar \nabla \Psi(\mathbf{r}) \,.$$

One occasionally even encounters an expression such as, though this is something of an abuse of notation. The differential operator must be understood to be an abstract operator, acting on kets, that has the effect of differentiating wavefunctions once the expression is projected onto the position basis, $$\nabla \lang\mathbf{r}|\Psi\rang \,,$$ even though, in the momentum basis, this operator amounts to a mere multiplication operator (by $|e_{k}\rangle$). That is, to say, $$ \langle \mathbf{r} |\hat \mathbf{p} = - i \hbar \nabla \langle \mathbf{r}| ~,$$ or $$ \hat \mathbf{p} = \int d^3 \mathbf{r} ~| \mathbf{r}\rangle ( - i \hbar \nabla) \langle \mathbf{r}| ~.$$

Overlap of states
In quantum mechanics the expression $ψ(x)$ is typically interpreted as the probability amplitude for the state $|ψ\rangle = ∫ dx ψ(x)|x\rangle$ to collapse into the state $x$. Mathematically, this means the coefficient for the projection of $|x\rangle$ onto ${ |r\rangle }$. It is also described as the projection of state $r$ onto state $|Ψ\rangle$.

Changing basis for a spin-1/2 particle
A stationary spin-1⁄2 particle has a two-dimensional Hilbert space. One orthonormal basis is: $$|{\uparrow}_z \rangle \,, \; |{\downarrow}_z \rangle$$ where $r$ is the state with a definite value of the spin operator $Ψ(r)$ equal to +1⁄2 and $iħp$ is the state with a definite value of the spin operator $⟨φ$ equal to −1⁄2.

Since these are a basis, any quantum state of the particle can be expressed as a linear combination (i.e., quantum superposition) of these two states: $$|\psi \rangle = a_{\psi} |{\uparrow}_z \rangle + b_{\psi} |{\downarrow}_z \rangle$$ where $ψ$ and $φ$ are complex numbers.

A different basis for the same Hilbert space is: $$|{\uparrow}_x \rangle \,, \; |{\downarrow}_x \rangle$$ defined in terms of $ψ$ rather than $φ$.

Again, any state of the particle can be expressed as a linear combination of these two: $$|\psi \rangle = c_{\psi} |{\uparrow}_x \rangle + d_{\psi} |{\downarrow}_x \rangle$$

In vector form, you might write $$|\psi\rangle \doteq \begin{pmatrix} a_\psi \\ b_\psi \end{pmatrix} \quad \text{or} \quad |\psi\rangle \doteq \begin{pmatrix} c_\psi \\ d_\psi \end{pmatrix} $$ depending on which basis you are using. In other words, the "coordinates" of a vector depend on the basis used.

There is a mathematical relationship between $$a_\psi$$, $$b_\psi$$, $$c_\psi$$ and $$d_\psi$$; see change of basis.

Pitfalls and ambiguous uses
There are some conventions and uses of notation that may be confusing or ambiguous for the non-initiated or early student.

Separation of inner product and vectors
A cause for confusion is that the notation does not separate the inner-product operation from the notation for a (bra) vector. If a (dual space) bra-vector is constructed as a linear combination of other bra-vectors (for instance when expressing it in some basis) the notation creates some ambiguity and hides mathematical details. We can compare bra–ket notation to using bold for vectors, such as $$\boldsymbol \psi$$, and $$(\cdot,\cdot)$$ for the inner product. Consider the following dual space bra-vector in the basis $$\{|e_n\rangle\}$$: $$\langle\psi| = \sum_n \langle e_n| \psi_n$$

It has to be determined by convention if the complex numbers $$\{\psi_n\}$$ are inside or outside of the inner product, and each convention gives different results.

$$\langle\psi| \equiv (\boldsymbol\psi, \cdot ) = \sum_n (\boldsymbol e_n, \cdot ) \, \psi_n$$ $$\langle\psi| \equiv (\boldsymbol\psi, \cdot ) = \sum_n (\boldsymbol e_n \psi_n, \cdot ) = \sum_n (\boldsymbol e_n, \cdot ) \, \psi_n^*$$

Reuse of symbols
It is common to use the same symbol for labels and constants. For example, $$\hat \alpha |\alpha\rangle = \alpha |\alpha \rangle$$, where the symbol $$\alpha$$ is used simultaneously as the name of the operator $$\hat \alpha$$, its eigenvector $$|\alpha\rangle$$ and the associated eigenvalue $$\alpha$$. Sometimes the hat is also dropped for operators, and one can see notation such as $$A |a\rangle = a |a \rangle$$.

Hermitian conjugate of kets
It is common to see the usage $$|\psi\rangle^\dagger = \langle\psi|$$, where the dagger ($$\dagger$$) corresponds to the Hermitian conjugate. This is however not correct in a technical sense, since the ket, $$|\psi\rangle$$, represents a vector in a complex Hilbert-space $$\mathcal{H}$$, and the bra, $$\langle\psi|$$, is a linear functional on vectors in $$\mathcal{H}$$. In other words, $$|\psi\rangle$$ is just a vector, while $$\langle\psi|$$ is the combination of a vector and an inner product.

Operations inside bras and kets
This is done for a fast notation of scaling vectors. For instance, if the vector $$|\alpha \rangle$$ is scaled by $$1/\sqrt{2}$$, it may be denoted $$|\alpha/\sqrt{2} \rangle$$. This can be ambiguous since $$\alpha$$ is simply a label for a state, and not a mathematical object on which operations can be performed. This usage is more common when denoting vectors as tensor products, where part of the labels are moved outside the designed slot, e.g. $$|\alpha \rangle = |\alpha/\sqrt{2} \rangle_1 \otimes |\alpha/\sqrt{2} \rangle_2$$.

Linear operators acting on kets
A linear operator is a map that inputs a ket and outputs a ket. (In order to be called "linear", it is required to have certain properties.) In other words, if $$\hat A$$ is a linear operator and $$|\psi\rangle$$ is a ket-vector, then $$\hat A |\psi\rangle$$ is another ket-vector.

In an $$N$$-dimensional Hilbert space, we can impose a basis on the space and represent $$|\psi\rangle$$ in terms of its coordinates as a $$N \times 1$$ column vector. Using the same basis for $$\hat A$$, it is represented by an $$N \times N$$ complex matrix. The ket-vector $$\hat A |\psi\rangle$$ can now be computed by matrix multiplication.

Linear operators are ubiquitous in the theory of quantum mechanics. For example, observable physical quantities are represented by self-adjoint operators, such as energy or momentum, whereas transformative processes are represented by unitary linear operators such as rotation or the progression of time.

Linear operators acting on bras
Operators can also be viewed as acting on bras from the right hand side. Specifically, if $ψ$ is a linear operator and $φ$ is a bra, then $|↑_{z}\rangle$ is another bra defined by the rule $$\bigl(\langle\phi|\boldsymbol{A}\bigr) |\psi\rangle = \langle\phi| \bigl(\boldsymbol{A}|\psi\rangle\bigr) \,,$$ (in other words, a function composition). This expression is commonly written as (cf. energy inner product) $$\langle\phi| \boldsymbol{A} |\psi\rangle \,.$$

In an $S_{z}$-dimensional Hilbert space, $|↓_{z}\rangle$ can be written as a $S_{z}$ row vector, and $a_{ψ}$ (as in the previous section) is an $b_{ψ}$ matrix. Then the bra $S_{x}$ can be computed by normal matrix multiplication.

If the same state vector appears on both bra and ket side, $$\langle\psi|\boldsymbol{A}|\psi\rangle \,,$$ then this expression gives the expectation value, or mean or average value, of the observable represented by operator $S_{z}$ for the physical system in the state $A$.

Outer products
A convenient way to define linear operators on a Hilbert space $\langleφ|$ is given by the outer product: if $\langleφ|A$ is a bra and $N$ is a ket, the outer product $$ |\phi\rang \, \lang \psi| $$ denotes the rank-one operator with the rule $$ \bigl(|\phi\rang \lang \psi|\bigr)(x) = \lang \psi | x \rang |\phi \rang.$$

For a finite-dimensional vector space, the outer product can be understood as simple matrix multiplication: $$ |\phi \rangle \, \langle \psi | \doteq \begin{pmatrix} \phi_1 \\ \phi_2 \\ \vdots \\ \phi_N \end{pmatrix} \begin{pmatrix} \psi_1^* & \psi_2^* & \cdots & \psi_N^* \end{pmatrix} = \begin{pmatrix} \phi_1 \psi_1^* & \phi_1 \psi_2^* & \cdots & \phi_1 \psi_N^* \\ \phi_2 \psi_1^* & \phi_2 \psi_2^* & \cdots & \phi_2 \psi_N^* \\ \vdots & \vdots & \ddots & \vdots \\ \phi_N \psi_1^* & \phi_N \psi_2^* & \cdots & \phi_N \psi_N^* \end{pmatrix} $$ The outer product is an $\langleφ|$ matrix, as expected for a linear operator.

One of the uses of the outer product is to construct projection operators. Given a ket $1 × N$ of norm 1, the orthogonal projection onto the subspace spanned by $A$ is $$|\psi\rangle \, \langle\psi| \,.$$ This is an idempotent in the algebra of observables that acts on the Hilbert space.

Hermitian conjugate operator
Just as kets and bras can be transformed into each other (making $N × N$ into $\langleφ|A$), the element from the dual space corresponding to $A$ is $|ψ\rangle$, where $\mathcal{H}$ denotes the Hermitian conjugate (or adjoint) of the operator $\langleϕ|$. In other words, $$ |\phi\rangle = A |\psi\rangle \quad \text{if and only if} \quad \langle\phi| = \langle \psi | A^\dagger \,.$$

If $|ψ\rangle$ is expressed as an $N × N$ matrix, then $|ψ\rangle$ is its conjugate transpose.

Self-adjoint operators, where $|ψ\rangle$, play an important role in quantum mechanics; for example, an observable is always described by a self-adjoint operator. If $A$ is a self-adjoint operator, then $|ψ\rangle$ is always a real number (not complex). This implies that expectation values of observables are real.

Properties
Bra–ket notation was designed to facilitate the formal manipulation of linear-algebraic expressions. Some of the properties that allow this manipulation are listed herein. In what follows, $\langleψ|$ and $A|ψ\rangle$ denote arbitrary complex numbers, $\langleψ|A^{†}$ denotes the complex conjugate of $A^{†}$, $A$ and $A$ denote arbitrary linear operators, and these properties are to hold for any choice of bras and kets.

Linearity

 * Since bras are linear functionals, $$\langle\phi| \bigl( c_1|\psi_1\rangle + c_2|\psi_2\rangle \bigr) = c_1\langle\phi|\psi_1\rangle + c_2\langle\phi|\psi_2\rangle \,. $$
 * By the definition of addition and scalar multiplication of linear functionals in the dual space, $$\bigl(c_1 \langle\phi_1| + c_2 \langle\phi_2|\bigr) |\psi\rangle = c_1 \langle\phi_1|\psi\rangle + c_2 \langle\phi_2|\psi\rangle \,. $$

Associativity
Given any expression involving complex numbers, bras, kets, inner products, outer products, and/or linear operators (but not addition), written in bra–ket notation, the parenthetical groupings do not matter (i.e., the associative property holds). For example:
 * $$\begin{align}

\lang \psi| \bigl(A |\phi\rang\bigr) = \bigl(\lang \psi|A\bigr)|\phi\rang \, &\stackrel{\text{def}}{=} \, \lang \psi | A | \phi \rang \\ \bigl(A|\psi\rang\bigr)\lang \phi| = A\bigl(|\psi\rang \lang \phi|\bigr) \, &\stackrel{\text{def}}{=} \, A | \psi \rang \lang \phi | \end{align}$$ and so forth. The expressions on the right (with no parentheses whatsoever) are allowed to be written unambiguously because of the equalities on the left. Note that the associative property does not hold for expressions that include nonlinear operators, such as the antilinear time reversal operator in physics.

Hermitian conjugation
Bra–ket notation makes it particularly easy to compute the Hermitian conjugate (also called dagger, and denoted $N × N$) of expressions. The formal rules are:
 * The Hermitian conjugate of a bra is the corresponding ket, and vice versa.
 * The Hermitian conjugate of a complex number is its complex conjugate.
 * The Hermitian conjugate of the Hermitian conjugate of anything (linear operators, bras, kets, numbers) is itself—i.e., $$\left(x^\dagger\right)^\dagger=x \,.$$
 * Given any combination of complex numbers, bras, kets, inner products, outer products, and/or linear operators, written in bra–ket notation, its Hermitian conjugate can be computed by reversing the order of the components, and taking the Hermitian conjugate of each.

These rules are sufficient to formally write the Hermitian conjugate of any such expression; some examples are as follows: \bigl(c_1|\psi_1\rangle + c_2|\psi_2\rangle\bigr)^\dagger = c_1^* \langle\psi_1| + c_2^* \langle\psi_2| \,. $$ \langle \phi| A | \psi \rangle^\dagger &= \left\langle \psi \left| A^\dagger \right|\phi \right\rangle \\ \left\langle \phi\left| A^\dagger B^\dagger \right| \psi \right\rangle^\dagger &= \langle \psi | BA |\phi \rangle \,. \end{align}$$
 * Kets: $$
 * Inner products: $$\langle \phi | \psi \rangle^* = \langle \psi|\phi\rangle \,.$$ Note that $A^{†}$ is a scalar, so the Hermitian conjugate is just the complex conjugate, i.e., $$\bigl(\langle \phi | \psi \rangle\bigr)^\dagger = \langle \phi | \psi \rangle^*$$
 * Matrix elements: $$\begin{align}
 * Outer products: $$\Big(\bigl(c_1|\phi_1\rangle\langle \psi_1|\bigr) + \bigl(c_2|\phi_2\rangle\langle\psi_2|\bigr)\Big)^\dagger = \bigl(c_1^* |\psi_1\rangle\langle \phi_1|\bigr) + \bigl(c_2^*|\psi_2\rangle\langle\phi_2|\bigr) \,.$$

Composite bras and kets
Two Hilbert spaces $A = A^{†}$ and $\langleψ|A|ψ\rangle$ may form a third space $c_{1}$ by a tensor product. In quantum mechanics, this is used for describing composite systems. If a system is composed of two subsystems described in $c_{2}$ and $c*$ respectively, then the Hilbert space of the entire system is the tensor product of the two spaces. (The exception to this is if the subsystems are actually identical particles. In that case, the situation is a little more complicated.)

If $c$ is a ket in $A$ and $B$ is a ket in $†$, the tensor product of the two kets is a ket in $⟨φ$. This is written in various notations:
 * $$|\psi\rangle|\phi\rangle \,,\quad |\psi\rangle \otimes |\phi\rangle\,,\quad|\psi \phi\rangle\,,\quad|\psi ,\phi\rangle\,.$$

See quantum entanglement and the EPR paradox for applications of this product.

The unit operator
Consider a complete orthonormal system (basis), $$\{ e_i \ | \ i \in \mathbb{N} \} \,,$$ for a Hilbert space $V$, with respect to the norm from an inner product $W$.

From basic functional analysis, it is known that any ket $$|\psi\rangle $$ can also be written as $$|\psi\rangle = \sum_{i \in \mathbb{N}} \langle e_i | \psi \rangle | e_i \rangle,$$ with $V ⊗ W$ the inner product on the Hilbert space.

From the commutativity of kets with (complex) scalars, it follows that $$\sum_{i \in \mathbb{N}} | e_i \rangle \langle e_i | = \mathbb{I}$$ must be the identity operator, which sends each vector to itself.

This, then, can be inserted in any expression without affecting its value; for example $$\begin{align} \langle v | w \rangle &= \langle v | \left( \sum_{i \in \mathbb{N}} | e_i \rangle \langle e_i| \right) | w \rangle \\ &= \langle v | \left( \sum_{i \in \mathbb{N}} | e_i \rangle \langle e_i| \right) \left( \sum_{j \in \mathbb{N}} | e_j \rangle \langle e_j |\right)| w \rangle \\ &= \langle v | e_i \rangle \langle e_i | e_j \rangle \langle e_j | w \rangle \,, \end{align}$$ where, in the last line, the Einstein summation convention has been used to avoid clutter.

In quantum mechanics, it often occurs that little or no information about the inner product $V$ of two arbitrary (state) kets is present, while it is still possible to say something about the expansion coefficients $W$ and $|ψ\rangle$ of those vectors with respect to a specific (orthonormalized) basis. In this case, it is particularly useful to insert the unit operator into the bracket one time or more.

For more information, see Resolution of the identity, $$ {\mathbb I} = \int\! dx~ | x \rangle \langle x |= \int\! dp ~| p \rangle \langle p |,$$ where $$|p\rangle = \int dx \frac{e^{ixp / \hbar} |x\rangle}{\sqrt{2\pi\hbar}}.$$

Since $V$, plane waves follow, $$ \langle x | p \rangle = \frac{e^{ixp / \hbar}}{\sqrt{2\pi\hbar}}.$$

In his book (1958), Ch. III.20, Dirac defines the standard ket which, up to a normalization, is the translationally invariant momentum eigenstate $|\varpi\rangle=\lim_{p\to 0} |p\rangle$ in the momentum representation, i.e., $$\hat{p}|\varpi\rangle=0$$. Consequently, the corresponding wavefunction is a constant, $$ \langle x|\varpi\rangle \sqrt{2\pi \hbar}=1$$, and $$|x\rangle= \delta(\hat{x}-x) |\varpi\rangle \sqrt{2\pi \hbar},$$ as well as $$|p\rangle= \exp (ip\hat{x}/\hbar ) |\varpi\rangle.$$

Typically, when all matrix elements of an operator such as $$ \langle x| A |y\rangle $$ are available, this resolution serves to reconstitute the full operator, $$ \int dx \, dy \, |x\rangle \langle x| A |y\rangle \langle y | = A \,. $$

Notation used by mathematicians
The object physicists are considering when using bra–ket notation is a Hilbert space (a complete inner product space).

Let $$(\mathcal H, \langle\cdot,\cdot\rangle)$$ be a Hilbert space and $|φ\rangle$ a vector in $W$. What physicists would denote by $V ⊗ W$ is the vector itself. That is, $$ |h\rangle\in \mathcal{H} .$$

Let $H$ be the dual space of $⟨·,·⟩$. This is the space of linear functionals on $⟨·$. The embedding $$\Phi:\mathcal H \hookrightarrow \mathcal H^*$$ is defined by $$\Phi(h) = \varphi_h$$, where for every $⟨ψ$ the linear functional $$\varphi_h:\mathcal H\to\mathbb C$$ satisfies for every $e_{i}⟩ = ⟨e_{i}$ the functional equation $$\varphi_h(g) = \langle h, g\rangle = \langle h\mid g\rangle$$. Notational confusion arises when identifying $⟨e_{i}$ and $x⟩ = δ(x − x)$ with $h ∈ \mathcal{H}$ and $\mathcal{H}$ respectively. This is because of literal symbolic substitutions. Let $$\varphi_h = H = \langle h\mid$$ and let $|h\rangle$. This gives $$ \varphi_h(g) = H(g) = H(G)=\langle h|(G) = \langle h|\bigl(|g\rangle\bigr) \,. $$

One ignores the parentheses and removes the double bars.

Moreover, mathematicians usually write the dual entity not at the first place, as the physicists do, but at the second one, and they usually use not an asterisk but an overline (which the physicists reserve for averages and the Dirac spinor adjoint) to denote complex conjugate numbers; i.e., for scalar products mathematicians usually write $$\langle\phi ,\psi\rangle=\int \phi (x)\cdot \overline{\psi(x)}\, \mathrm{d}x \,,$$ whereas physicists would write for the same quantity $$ \langle\psi |\phi \rangle = \int dx \, \psi^*(x) \phi(x)~.$$

title: Bra–ket notation