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Galusha A. Grow

Galusha Aaron Grow (August 31, 1823 – March 31, 1907) was an American politician, lawyer, writer and businessman, who served as 24th Speaker of the U.S. House of Representatives from 1861 to 1863. Elected as a Democrat in the 1850 congressional elections, he switched to the newly organized Republican Party in the mid-1850s when the Democratic Party tried to force the extension of slavery into western territories.

Elected speaker for the 37th Congress, Grow presided over the House during the initial years of the American Civil War. During his tenure Congress passed the landmark Homestead Act of 1862, which he supported. Grow was defeated for reelection in 1862. For over a century he remained the last incumbent House speaker to be defeated, until Speaker Tom Foley lost his seat in 1994.

After leaving office he continued to speak out on political issues, but did not serve in elective office. Then, 31 years after leaving office, Grow won an 1894 special election to succeed William Lilly. It remains one of the longest known interregnums between terms of service for a House member. Over the course of his career, Grow represented the people of three Pennsylvania congressional districts: the PA's 12th congressional district (1851–1853), PA's 14th congressional district (1853–1863), and PA's at-large congressional district (1894–1903).

Early life and education
Grow was born Aaron Galusha Grow in Ashford, Connecticut. His given names were the suggestions of an aunt living in Vermont, who was visiting Grow's mother when he was christened: "Aaron" was the aunt's husband's name (his full name was Aaron Nichols (1764–1807)), and "Galusha" was the surname of a governor of Vermont she admired. His family called him Galusha when he was growing up, and before Grow was a teenager, he had started writing his name with his given names reversed. He was educated at Franklin Academy in Susquehanna County, Pennsylvania, and later at Amherst College. He studied law and was admitted to the bar in November 1847 and then began his law practice.

Early elections
Grow ran as a Democrat in the 1850 election and served as a member of that party during the 32nd and 33rd congresses, and into the 34th Congress. He switched parties in the wake of President Pierce's signing of the Kansas-Nebraska Act. He ran as a Republican in the 1856 election and remained a member of that party for the rest of his political career.

Congressional brawl of 1858
During the 35th United States Congress, on February 5, 1858, he was physically attacked by Democrat Laurence M. Keitt in the House chambers, leading to a brawl between northerners and southerners. Keitt, offended by Grow's having stepped over to his side of the House chamber, dismissively demanded that Grow sit down, calling him a "black Republican puppy". Grow responded by telling Keitt that "No negro-driver shall crack his whip over me." Keitt became enraged and went for Grow's throat, shouting that he would "choke [him] for that". A large brawl involving approximately fifty representatives erupted on the House floor, ending only when a missed punch from Rep. Cadwallader Washburn upended the hairpiece of Rep. William Barksdale. The embarrassed Barksdale accidentally replaced the wig backwards, causing both sides to erupt in spontaneous laughter.

Later that year Grow was re-elected to a fifth term. When the next Congress convened in December 1859, he was one of 90 congressmen to receive votes during the two-month-long 44-ballot speaker election, dropping out following the first ballot. On December 29, 1859, North Carolina Congressman Lawrence O'Bryan Branch challenged Galusha Grow to a duel after the two exchanged insults on the House Floor. Both men and their seconds were arrested by District of Columbia police before the duel could take place.

1861 speaker election
The deepening rift between slave states and free states overshadowed Grow's 1861 re-election victory, as a national crisis erupted in December 1860 when South Carolina became the first of several Southern states to adopt an Ordinance of Secession. Four months later, on April 12, 1861, Confederate forces fired upon Fort Sumter, igniting the Civil War. In response, President Abraham Lincoln called the 37th Congress into session on July 4. When the House convened that day, Grow was nominated to be Speaker of the House; also nominated was Francis Preston Blair Jr. Grow was elected on the first ballot, but only after Blair withdrew following the roll call vote, at which time 28 votes shifted to Grow.

1861 election for speaker – 159 votes cast, 80 votes necessary to win election: Galusha A. Grow (R–Pennsylvania) 99 Francis Preston Blair Jr. (R–Missouri) 12 John J. Crittenden (CU–Kentucky) 12 John S. Phelps (D–Missouri) 7 Clement Vallandingham (D–Ohio) 7 Erastus Corning (D–New York) 7 Samuel S. Cox (D–Ohio) 6 Others 9

Although events of the war dominated and the First Battle of Bull Run occurred only two weeks after the 37th Congress was called into session, under Grow's leadership, several major acts of Congress were passed and signed into law, particularly the Morrill Land-Grant College Act (passed House June 17, 1862), the Pacific Railway Act authorizing land grants to encourage the construction of the transcontinental railroad, and the Homestead Act, which in over a century resulted in the establishment of 1.6 million homesteads.

Loss of congressional seat and interim
Grow, a supporter of the Radical Republicans, was defeated in his re-election bid in 1862, becoming the second sitting House Speaker in a row to lose his seat.

Grow was a delegate to the Republican National Convention in 1864 and 1868.

He moved to Houston, Texas in 1871, and that year became president of what became known as the International - Great Northern Railroad, a position he held until 1875. He then returned to Pennsylvania and the practice of law from 1875 to 1894.

Return to Congress
Grow returned to the United States Congress as a member at-large from Pennsylvania from 1894 to 1903; was the chairman of the committee on education in the 56th Congress.

Death and legacy
Grow resided in Glenwood, Pennsylvania, from 1903 until his death at age 84. A biography of Galusha Grow, Galusha A. Grow: Father of the Homestead Law, was written by James T. Du Bois and Gertrude S. Mathews and published by Houghton Mifflin in 1917.

A monument to Grow was erected in 1915 at the Susquehanna County Courthouse Complex in Montrose, Pennsylvania.

title: Galusha A. Grow


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Chaquan

Chaquan is a Chinese martial art that features graceful movements and some acrobatic aerial maneuvers and includes a large range of weapons. Chaquan falls under the classification Changquan, a type of Northern Chinese martial arts known for their extended, long movements.

The style is associated with the Hui people and related to the Turkic people from Central Asia. In the legend, a Turkic warrior named "Zha Mi-Er" (maybe Sameer [name of Arabic origin] or Dämir [meaning "iron" in Turkic]); ) from current Xinjiang or Central Asia passed down this martial art to the Chinese locals in the current Xandong province during the late Ming dynasty. One famous master of Chaquan was Wang Ziping, who was known for his great strength. Other modern day masters include Zhang Wenguang, Ma Jinbiao, and Liu Hongchi.

Chaquan is one of the sources of the contemporary Wushu Changquan, which is often seen in movies and tournaments.

Chaquan is a system that has six main weapons (staff, saber, sword, spear, kwandao, hookswords). It emphasizes long range movements and stances combined with speed and power. The style includes many forms, including 10 lines of tantui for basic power training, 10 longer sets of Chaquan, and other forms as well.

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

title: Chaquan


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User talk:Bosniak

Proposed deletion of Srebrenica Children Massacre

The article Srebrenica Children Massacre has been proposed for deletion  because of the following concern:
 * None of the article's sources are available, about this massacre, or describe it by this name. Thus proposing to delete as unsourced and/or non-notable.

While all contributions to Wikipedia are appreciated, content or articles may be deleted for any of several reasons.

You may prevent the proposed deletion by removing the  notice, but please explain why in your edit summary or on the article's talk page.

Please consider improving the article to address the issues raised. Removing  will stop the proposed deletion process, but other deletion processes exist. The speedy deletion process can result in deletion without discussion, and articles for deletion allows discussion to reach consensus for deletion. Psychonaut (talk) 14:20, 22 January 2010 (UTC)

Y'''ou dirty piece of Serbian shit. Now that you blocked me, now you want to delete articles dealing with Serb crimes? All sources are valid with exception to UN site moving to ICTY.org. Oh, you dirty Serbian bastards. Bosniak (talk) 07:06, 24 January 2010 (UTC) '''

I have other IP addresses with different accounts. Anyways, dirty serbian bastards, you accomplished nothing.

Note: I've protected this talk page as a result of the above abuse, but leave the unblock request visible for another administrator to formally process.  Sandstein  07:32, 24 January 2010 (UTC)

title: User talk:Bosniak


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Voiced velar nasal

The voiced velar nasal, also known as eng, engma, or agma (from Greek ἆγμα âgma 'fragment'), is a type of consonantal sound used in some spoken languages. It is the sound of ng in English sing as well as n before velar consonants as in English and ink. The symbol in the International Phonetic Alphabet that represents this sound is ⟨ŋ⟩, and the equivalent X-SAMPA symbol is. The IPA symbol ⟨ŋ⟩ is similar to ⟨ɳ⟩, the symbol for the retroflex nasal, which has a rightward-pointing hook extending from the bottom of the right stem, and to ⟨ɲ⟩, the symbol for the palatal nasal, which has a leftward-pointing hook extending from the bottom of the left stem.

While almost all languages have and  as phonemes,  is rarer. Half of the 469 languages surveyed in had a velar nasal phoneme; as a further curiosity, many of them limit its occurrence to the syllable coda. The velar nasal does not occur in many of the languages of the Americas, the Middle East, or the Caucasus, but it is extremely common among Australian Aboriginal languages, languages of Sub-Saharan Africa, East Asian and Southeast Asian languages, and Polynesian languages. In many languages that do not have the velar nasal as a phoneme, such as the Romance languages, it occurs as an allophone of before velar consonants. This kind of assimilation can even be found in languages with phonemic voiced velar nasals, such as English. An example of this phenomenon is the word income; its underlying representation,, can be realized as either or.

An example of a language that lacks a phonemic or allophonic velar nasal is Russian, in which is pronounced as laminal denti-alveolar  even before velar consonants.

Some languages have the pre-velar nasal, which is articulated slightly more front compared with the place of articulation of the prototypical velar nasal, though not as front as the prototypical palatal nasal - see that article for more information.

Conversely, some languages have the post-velar nasal, which is articulated slightly behind the place of articulation of a prototypical velar nasal, though not as back as the prototypical uvular nasal.

Features

Features of the voiced velar nasal:

title: Voiced velar nasal


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Neil Sedaka

Neil Sedaka (born March 13, 1939) is an American singer, songwriter and pianist. Since his music career began in 1957, he has sold millions of records worldwide and has written or co-written over 500 songs for himself and other artists, collaborating mostly with lyricists Howard "Howie" Greenfield and Phil Cody.

After a short-lived tenure as a founding member of the doo-wop group the Tokens, Sedaka achieved a string of hit singles over the late 1950s and early 1960s, including "Oh! Carol" (1959), "Calendar Girl" (1960), "Happy Birthday Sweet Sixteen" (1961) and "Breaking Up Is Hard to Do" (1962). His popularity declined by the mid-1960s, but was revived in the mid-1970s, solidified by the 1975 US Billboard Hot 100 number ones "Laughter in the Rain" and "Bad Blood". Sedaka maintained a successful career as a songwriter, penning hits for other artists including "Stupid Cupid" (Connie Francis), "(Is This the Way to) Amarillo" (Tony Christie) and "Love Will Keep Us Together" (Captain & Tennille). He was inducted into the Songwriters Hall of Fame in 1983 and continues to perform, mounting mini-concerts on social media during the COVID-19 pandemic.

Early life: Juilliard and the Brill Building
Sedaka was born in Brooklyn, New York. His father, Mordechai "Mac" Sedaka, was a taxi driver of Lebanese Jewish descent. Sedaka's paternal grandparents came to the United States from Istanbul in 1910. Sedaka's mother, Eleanor (née Appel), was an Ashkenazi Jew of Polish and Russian descent. He grew up in Brighton Beach. His father's cousin, Rachel Gorman (née Cohen), daughter of Isaac Cohen and Calo Cohen (née Sedaca or Sedaka), was married to Morris Gorman (né Garmezano; paternal uncle to singer Eydie Gormé). Gormé's mother was of Lebanese Jewish descent.

Sedaka demonstrated musical aptitude in his second-grade choral class and, when his teacher sent a note home suggesting he take piano lessons, his mother took a part-time job in an Abraham & Straus department store for six months to pay for a second-hand upright. In 1947, he auditioned successfully for a piano scholarship to the Juilliard School of Music's Preparatory Division for Children, which he attended on Saturdays. His mother had wanted him to become a classical pianist like his contemporary Van Cliburn, and Sedaka continued to show fondness for (and capacity to play) classical music throughout his life.

At the same time, to his mother's dismay, Sedaka was discovering pop music; his mother eventually acquiesced when Sedaka received a five-figure royalty check for his hit "Calendar Girl" in 1961. When Sedaka was 13, a neighbor heard him playing and introduced him to her 16-year-old son, Howard Greenfield, an aspiring poet and lyricist. They became two of the Brill Building's composers.

Sedaka and Greenfield wrote songs together throughout much of their young lives. Before rock and roll became popular, Sedaka and Greenfield found inspiration from show tunes. When Sedaka became a major teen pop star, the pair continued writing hits for Sedaka and numerous other artists. When the Beatles and the British Invasion took American music in a different direction, Sedaka was left without a recording career. In the early 1970s, he decided a major change in his life was necessary and moved his family to Britain. Sedaka and Greenfield mutually agreed to end their partnership with "Our Last Song Together". Sedaka began a new composing partnership with lyricist Phil Cody, from Pleasantville, New York.

Rise to fame with RCA Victor: the late 1950s
After graduating from Abraham Lincoln High School, Sedaka and some of his classmates formed a band called the Linc-Tones. The band had minor regional hits with songs like "While I Dream", "I Love My Baby", "Come Back, Joe", and "Don't Go", before Sedaka launched his solo career and left the group in 1957. The Linc-Tones, later renamed the Tokens near the end of Sedaka's tenure with the group, went on to have four top-40 hits of their own without Sedaka. Sedaka's first three solo singles, "Laura Lee", "Ring-a-Rockin'", and "Oh, Delilah!" failed to become hits (although "Ring-a-Rockin'" earned him the first of many appearances on Dick Clark's American Bandstand), but they demonstrated his ability to perform as a solo singer, so RCA Victor signed him to a recording contract.

His first single for RCA Victor, "The Diary", was inspired by Connie Francis, one of Sedaka and Greenfield's most important clients, while the three were taking a temporary break during their idea-making for a new song. Francis was writing in her diary, Sedaka asked if he could read it, and Connie said no. After Little Anthony and the Imperials passed on the song, Sedaka recorded it himself, and his debut single hit the Top 15 on the Billboard Hot 100, peaking at No. 14 in 1958.

His second single, a novelty tune titled "I Go Ape", just missed the Top 40, peaking at No. 42, but it became a more successful single in the United Kingdom with a No. 9. The third single, "Crying My Heart Out for You", was a commercial failure, missing the Hot 100 entirely, peaking at No. 111 but reaching No. 6 on the pop charts in Italy. RCA Victor had lost money on "I Go Ape" and "Crying My Heart Out For You" and was ready to drop Sedaka from their label; Sedaka feared he was headed for one-hit wonder status. Sedaka and his manager, Al Nevins, persuaded the RCA executives to give him one more chance.

Sedaka then bought the three biggest hit singles of the time and listened to them repeatedly, studying the song structure, chord progressions, lyrics and harmonies before writing his next songs. "Oh! Carol" delivered Sedaka his first domestic Top 10 hit, reaching No. 9 on the Hot 100 in 1959 and going to No. 1 on the Italian pop charts in 1960, giving Sedaka his first No. 1 ranking. In the UK, the song spent a total of 17 weeks in the top 40, peaking at No. 3 (4 weeks). In addition, the B-side, "One Way Ticket", reached No. 1 on the pop charts in Japan. Sedaka had dated Carole King when he was still at high school, which gave him the idea to use her name in the song. Gerry Goffin—King's husband—took the tune and wrote the playful response "Oh! Neil", which King recorded and released as an unsuccessful single the same year. Thus, this was the only time the melody of the song was used by a popular artist and a future sensation around the same time.

Big hits in the early 1960s
After establishing himself in 1958, Sedaka wrote many more hits from 1960 to 1962. His flow of Top 30 hits during this period included: "Stairway to Heaven" (No. 9, 1960); "You Mean Everything to Me" (No. 17, 1960); "Run, Samson, Run" (No. 27, 1960); "Calendar Girl" (No. 4, 1961; also reached No. 1 on the Japanese and Canadian pop charts); "Little Devil" (No. 11, 1961); "Happy Birthday Sweet Sixteen" (No. 6, 1961); his signature song, "Breaking Up Is Hard To Do" (No. 1, two weeks: August 11 and 18, 1962); and "Next Door to an Angel" (No. 5, 1962). For several of those songs, Sedaka was paired with Stan Applebaum and His Orchestra. Singles not making the Top 30 during this period included "Sweet Little You" (No. 59, 1961) and "King of Clowns" (No. 45, 1962). RCA Victor issued four LPs of his works in the United States and Great Britain during this period, and also produced Scopitone and Cinebox videos of "Calendar Girl" in 1961, "Breaking Up Is Hard To Do" in 1962, and "The Dreamer" in 1963. His second LP was an album mostly of old standards. He made regular appearances on such TV programs as American Bandstand and Shindig! during this period.

Connie Francis
When Sedaka was not recording his own songs, he and Howard Greenfield were writing for other performers, most notably in their earliest days Connie Francis. Francis began searching for a new hit after her 1958 single "Who's Sorry Now?". She was introduced to Sedaka and Greenfield, who played for her every ballad they had written. Francis began writing in her diary while the two played the last of their songs. After they finished, Francis told them they wrote beautiful ballads but that they were too intellectual for the young generation. Greenfield suggested that they play a song they had written for the Shepherd Sisters. Sedaka protested that Francis would be insulted by being played such a puerile song, but Greenfield reminded him Francis had not accepted their other suggestions and they had nothing to lose. After Sedaka played "Stupid Cupid", Francis told them they had just played her new hit. Francis' rendition of the song reached No. 14 on the Billboard charts, while it topped the UK Singles Chart.

While Francis was writing in her diary, Sedaka asked her if he could read what she had written. Francis said no. This inspired Sedaka to write "The Diary", his own first hit single. Sedaka and Greenfield wrote many of Connie Francis' hits, such as "Fallin'" and the "Theme from Where the Boys Are", the film in which she starred. This hit the Top 5 on the Billboard pop singles chart and Francis had several No. 1 singles. "Where the Boys Are" eventually became her signature song.

Jimmy Clanton
Sedaka and Greenfield also wrote some of Jimmy Clanton's hits, such as "Another Sleepless Night", "What Am I Gonna Do?" and "All the Words in the World". Sedaka himself recorded each of these three songs: "Another Sleepless Night" appears on his Rock With Sedaka debut album; "What Am I Gonna Do?" was the B-side of "Going Home to Mary Lou" and appeared on his 1961 album Neil Sedaka Sings "Little Devil" and His Other Hits; and "All the Words in the World" was recorded but was kept in the RCA Victor vaults until 1977, at the height of Sedaka's return to popularity, when it was released on the album Neil Sedaka: The '50s and '60s.

Session work
Sedaka also contributed some work as a session pianist during his heyday as a pop singer-songwriter. His piano playing is heard on "Dream Lover", Bobby Darin's 1959 hit.

Foreign-language recordings
Sedaka was very popular in Italy. Many of his English-language records were released there and proved quite successful, especially "Crying My Heart Out for You" (Italian No. 6, 1959) and "Oh! Carol" (Italian No. 1, 1960).

In 1961, Sedaka began to record some of his hits in Italian, starting with "Esagerata" and "Un giorno inutile", local versions of "Little Devil" and "I Must Be Dreaming", respectively. Other recordings followed, such as "Tu non lo sai" ("Breaking Up Is Hard to Do"), "Il re dei pagliacci" ("King of Clowns"), "I tuoi capricci" ("Look Inside Your Heart"), and "La terza luna" ("Waiting For Never"). "La terza luna" reached No. 1 on the Italian pop charts in April 1963. Cinebox videos exist for "La terza luna" and "I tuoi capricci". Sedaka's Italian diction was impressive; his recordings in Italian had very little American accent. RCA Victor's Italiana branch distributed his records in Italy and released three compilation LPs of Sedaka's Italian recordings.

Sedaka also recorded an album in Yiddish (Brighton Beach Memories – Neil Sedaka Sings Yiddish), several songs in Spanish including "Mi Vecinita" ("Next Door to an Angel"), a handful of songs in German, and one single apiece in Hebrew, Japanese, and Canadian French. His English-language recordings were also quite popular internationally; "One-Way Ticket to the Blues" and "Calendar Girl" reached No. 1 on the Japanese pop charts in 1959 and 1961. He also enjoyed popularity in Latin America for his Spanish-language recordings. Many of these were pressed onto 78 rpm discs.

Sedaka stated in 2020 that he was talked into focusing on the international markets because Elvis Presley, the biggest rock star in America, never toured overseas (largely because of Colonel Tom Parker's desire to avoid immigration officials), and because his publishers and managers considered it a much lower risk for a new performer not to face American audiences who knew him.

Mid-1960s
The year 1962 was one of the most important of Sedaka's career, with "Breaking Up Is Hard To Do" reaching No. 1 and "Next Door to an Angel" reaching No. 5. After this his popularity began to wane and his 1963 singles enjoyed only moderate success: "Alice In Wonderland" (No. 17), "Let's Go Steady Again" (No. 26), "The Dreamer" (No. 47), and "Bad Girl" (No. 33). "Bad Girl" was Sedaka's last Top 40 hit in the U.S. until 1974.

In 1964 Sedaka's career began a sharp decline, hastened by The Beatles' arrival on the radio and TV, and the rest of the so-called British Invasion. When describing the Beatles' effect on his career in the mid-1960s, Sedaka put it brusquely: "The Beatles—not good!" He would later describe it in 2023: "after five years of hits... the bottom fell out." From 1964 to 1966, only three of his singles cracked the Hot 100: "Sunny" (No. 86, 1964), "The World through a Tear" (No. 76, 1965), and "The Answer to My Prayer" (No. 89, 1965). His other singles from this era—"The Closest Thing To Heaven", "I Hope He Breaks Your Heart", "Let The People Talk", "The Answer Lies Within" and "We Can Make It If We Try"—all missed the Hot 100, the same fate since Sedaka's third U.S. single for RCA Victor, and became commercial failures.

To make matters worse, RCA Victor refused to release his new recording, "It Hurts to Be in Love", because he had not recorded at their own studios, as stipulated by his contract. Sedaka attempted another recording of this song in RCA's studios, but the results were unsatisfactory. Howard Greenfield and Helen Miller, the song's co-writers, offered it instead to Gene Pitney. Pitney took the existing musical track, replacing Sedaka's lead vocal track with his own. Everything else was Sedaka, including his own arrangement and backing vocals, piano-playing, and usual female backup singers. Pitney ended up with a No. 7 hit for himself and his record label, Musicor, in 1964.

For the remainder of his tenure with RCA Victor, Sedaka never fully recovered from the effects of Beatlemania, the loss of "It Hurts to Be in Love" to Pitney, or the failure of his recordings. RCA decided not to renew his contract when it expired in 1966, leaving Sedaka without a recording label. He went into retirement as a performing artist.

Although Sedaka's stature as a recording artist was at a low ebb in the late 1960s, he was able to maintain his career through songwriting. Because his publisher, Aldon Music, was acquired by Screen Gems, two of his songs were recorded by The Monkees. Other hits Sedaka wrote in this period included The Cyrkle's versions of "We Had a Good Thing Goin'" and "Workin' On a Groovy Thing"; a Top 40 R&B hit for Patti Drew in 1968; and a Top 20 pop hit for The 5th Dimension in 1969. Also, "Make the Music Play" was included on Frankie Valli's charting album Timeless.

On a 1965 episode of the quiz show I've Got a Secret, Sedaka's secret was that he was to represent the United States at the 1966 Tchaikovsky classical piano competition in Moscow. Unaware of Sedaka's secret, panelist Henry Morgan challenged Sedaka with the fact that the Soviet bureaucracy had outlawed rock 'n' roll music, and that any Western music young Russians wanted had to be smuggled into the country. Once Sedaka's secret had been revealed, he impressed the show's panelists with his performance of Frederic Chopin's "Fantaisie Impromptu". Morgan's warning turned out to be prescient, however: despite Sedaka's classical roots, his "other" life as a pop star spurred the Soviet Union to disqualify him from entering the competition.

Sedaka also made an appearance in the 1968 movie Playgirl Killer, where he performed a song called "The Waterbug".

Australia years
Sedaka worked to revive his solo career in the early 1970s. Despite his waning chart appeal in the US in the late 1960s, he remained very popular as a concert attraction, notably in the UK and Australia. In 2010, as a guest on Australian disc jockey Bob Rogers' radio show, he thanked Rogers and Australian music fans for standing by him during that challenging time: "You know, Bob, in my lean years—I called them 'The Hungry Years'—it was Bob Rogers and Australia who welcomed me." Sedaka made several trips to Australia to play cabaret dates, and his commercial comeback began when the single "Star-Crossed Lovers" became a major hit there. The song went to No. 1 nationally in April 1969—giving Sedaka his first charting single anywhere in four years. It also came in at No. 5 in Go-Set magazine's list of the Top 40 Australian singles of 1969. Later that year, with the support of Festival Records, he recorded a new LP of original material entitled Workin' on a Groovy Thing (released in the United Kingdom as Sounds of Sedaka) at Festival Studios in Sydney. It was co-produced by Festival staff producer Pat Aulton, with arrangements by John Farrar (who later achieved international fame for his work with Olivia Newton-John) and backing by Australian session musicians including guitarist Jimmy Doyle (Ayers Rock) and noted jazz musician-composer John Sangster. One of the tracks from the album, "Wheeling, West Virginia", reached No. 20 in Australia in early 1970. The album is also notable because it was Sedaka's first album to include collaborations with writers other than long-time lyricist Howard Greenfield; the title track featured lyrics by Roger Atkins and four other songs were co-written with Carole Bayer Sager.

Emergence and Solitaire
In 1971, Sedaka reunited with RCA and released the Emergence album. Singles from that album included "I'm A Song (Sing Me)", "Silent Movies", "Superbird", and "Rosemary Blue". Good friend and New York music impresario Don Kirshner attempted to make the U.S. release of "Emergence" a comeback for Sedaka, but the album and single releases had no appreciable success, and RCA showed little interest in promoting the album. After the failure of "Emergence" in the US market, Sedaka left New York and moved to the UK.

In 1972, Sedaka embarked on a successful British tour and was introduced by Harvey Lisberg to the four future members of 10cc (best known to American audiences for "I'm Not in Love" and "The Things We Do for Love") with whom he recorded the Solitaire album at their Strawberry Studios in Stockport issued by RCA in 1972. As well as the title track, "Solitaire", which was successfully covered by Andy Williams (UK Top 5 singles chart) and the Carpenters (US Top 20), it included two UK Top 40 singles, one of which ("Beautiful You") also charted briefly in America, Sedaka's first US chart appearance in ten years. "Solitaire" was the first collaboration between Sedaka and Phil Cody, whom Sedaka felt was an ideal lyricist for writing music in the singer-songwriter period; Sedaka and Cody "clicked" as songwriters and began a prolonged collaboration between the two. Sedaka took a different approach to songwriting with Cody compared to Greenfield, with whom he was arguing frequently near the end of their collaboration; whereas he allowed Greenfield in the studio during recording, he met with Cody off-site to write songs. Cody did not consider Sedaka to be a personal friend; he mainly saw his work with Sedaka as a work project to bring the singer into the modern era.

Newfound success
A year later he reconvened with the Strawberry team, who had by then charted with their own debut 10cc album, to record The Tra-La Days Are Over for MGM Records, which started the second phase of his career and included his original version of the hit song "Love Will Keep Us Together" (also a US No. 1 hit two years later for Captain & Tennille). This album also marked the effective end of his writing partnership with Greenfield, commemorated by the track "Our Last Song Together" (later the last hit song for Bo Donaldson and the Heywoods, whose version hit No. 95). They reunited, however, and composed together again, before Greenfield's death in 1986. From 1974 onward, Sedaka's records were issued in Europe and around the world on the Polydor label. His first album of new material with Polydor was Laughter in the Rain (1974).

Darby Crash referred to Sedaka as "the real godfather of punk" due to his unaffected, simple chord progressions and unpretentious lyrics, saying that Calendar Girl "had all of punk right there-you just had to get high and play that pure rock and roll just like that."

Career with The Rocket Record Company
Elton John and Sedaka met at a party in London in 1973. When John learned Sedaka had no American record label, he suggested Sedaka sign with his Rocket Record Company, Limited, and Sedaka accepted the proposition. When John visited Sedaka at his London apartment, they discussed plans for relaunching his career in the United States.

John said he had "always been a Sedaka fan anyway". He went on to say:

"So the basic plan was as simple as finding out what he wanted to have on his album—which turned out to be a compilation from his British albums. It had been like Elvis coming up and giving us the chance to release his records. We couldn't believe our luck."

Sedaka's Back
Sedaka returned to the U.S. album charts with the release of Sedaka's Back, a compilation of songs from three albums he had already recorded in the UK—namely "Solitaire", "The Tra-La Days Are Over", and "Laughter in the Rain". It was only the second Sedaka album ever to chart in the U.S. Sedaka was known principally as a singles artist up to that point in his career; his only other American charting album was Neil Sedaka Sings His Greatest Hits, a compilation of his early singles. Although the single was released in the autumn of 1974 and was very slow in building in sales and at radio, eventually Sedaka found himself once again topping the Billboard Hot 100 singles chart (February 1, 1975) with "Laughter in the Rain". It was Sedaka's second No. 1 single thus far at that point in his career (after 1962's original version of "Breaking Up Is Hard to Do") and solidly reestablished Sedaka's popularity in America. Sedaka followed the success of "Laughter in the Rain" with a more politically motivated song, "The Immigrant", inspired by the songwriters' parents and by John Lennon, then facing immigration issues at the time. "The Immigrant" reached number 22 but topped the easy listening charts.

Writing for artists of the 1970s
In late 1972, producer Stig Anderson approached Sedaka to write the lyrics for a single by a new Swedish pop quartet then known as Björn & Benny, Agnetha & Anni-Frid. Sedaka agreed, on the condition he liked the song. Anderson, who had co-written the Swedish original with lyricist Björn Ulvaeus and composer Benny Andersson, intended to enter "Ring Ring" in the 1973 Eurovision Song Contest and believed with strong English lyrics it had the potential to become an international hit. He sent a tape of the song together with a rough translation to Sedaka, who within days returned original lyrics, co-written with Phil Cody. The song was entered into the Swedish Eurovision selections on February 10, 1973, but placed third. The band, renamed ABBA, made "Ring Ring" the title track of their first album, released on March 26, 1973. The single, credited to Andersson, Ulvaeus, Anderson, Sedaka and Cody, reached number 1 in Sweden and Belgium, and charted in the top 5 in at least four other countries. Sedaka later said that ABBA's "songwriting and production are in a class by themselves."

Sedaka and Greenfield co-wrote "Love Will Keep Us Together", a No. 1 hit for Captain & Tennille and the biggest hit for the entire year of 1975. Toni Tennille paid tribute to Sedaka's welcome return to music business success with her ad lib of "Sedaka is back" in the outro while she was laying down her own backing vocals for the track. "Captain" Daryl Dragon and Toni also recorded a Spanish-language version of the song the same year that cracked the top half of Billboard's Hot 100 chart ("Por Amor Viviremos", US pop No. 49).

Overnight Success/The Hungry Years
In late 1975, Sedaka's most successful year of his career continued as he earned yet more chart success with the release of his second Rocket Records album, The Hungry Years. This album was an American edition of Sedaka's British Polydor album Overnight Success. The first single, "Bad Blood", hit No. 1 on the Billboard 100 and stayed there for three weeks (October 11, 18 and 25, 1975), was certified Gold by the Recording Industry Association of America (RIAA), and was the most commercially successful individual single of his career. Elton John provided uncredited backing vocals for "Bad Blood". Despite their later falling out that resulted in Sedaka moving from Elton's Rocket Records to Elektra, Sedaka has credited John as being responsible for his successful return to the U.S. pop music scene. John has stated, "I only appear on the records of people I really know or like."

Another highlight from The Hungry Years was Sedaka's new version of "Breaking Up Is Hard to Do". His 1962 original, a No. 1 hit single, was upbeat; the remake was a slow ballad, which Sedaka had arranged for Lenny Welch five years prior. Sedaka's version hit No. 8 on the Hot 100 in early 1976, making him the only artist to ever record an entirely reinterpreted version of a song where both versions reached the Billboard Top 10. (Welch's version had reached No. 34.) The 1976 ballad version also hit No. 1 on Billboard's Adult Contemporary chart.

Steppin' Out
Later in 1976, Sedaka released a third and final album with Elton John's label The Rocket Record Company, Steppin' Out. The first single, "Love in the Shadows," was an uncharacteristically solid rock song featuring a scorching guitar solo. While it peaked at No. 16 on the Hot 100, it was the first of his three "comeback" albums' debut singles not to hit No. 1—or even the Top 10. The second single was the album's title track, once again featuring Elton on uncredited backing vocals. While it cracked the Top 40 (peaking at No. 36), it also marked the beginning of a slowdown in Sedaka's music sales and radio play, not unlike what he experienced in 1964 when The Beatles and the British Invasion arrived.

Sedaka subsequently left Rocket and signed with Elektra Records.

Sedaka met John again several times after his departure from Rocket, and he described their meetings as "cordial, but cold". The ice eventually thawed, however, and in the foreword to Sedaka's 2013 biography, John wrote of their friendship in glowing and positive terms.

Transition from Rocket to Elektra
Sedaka recorded four albums for Elektra Records: A Song (which produced the adult contemporary hits "Alone at Last" and a recording of "Amarillo"), All You Need Is the Music, In the Pocket and Neil Sedaka Now. The only top-40 pop hit to emerge from the Elektra records, as well as Sedaka's last to date, was "Should've Never Let You Go", a duet between Sedaka and his daughter Dara, which had appeared on In the Pocket in 1980.

Reissue of RCA-era recordings
Throughout the 1970s, Sedaka's former record company, RCA, reissued his 1960s-era songs on compilation LPs on the RCA Victor and RCA Camden labels. Sedaka released a final album of new material with RCA, consisting of a live concert he gave in Sydney. The album was released on the RCA International label in Australia and Europe as Neil Sedaka On Stage in 1974.

1980s and 1990s
Mac Sedaka, Neil's father, died on June 6, 1981, of metastatic colon cancer; Neil was at his bedside singing his father's favorite song, Sedaka's 1965 song "Pictures From The Past", when his father briefly awoke from his coma before his death. Sedaka re-recorded the song that same year.

Sedaka left Elektra and signed with Curb Records. Sedaka recorded two albums on the Curb label – Come See About Me, a covers album, in 1983 and The Good Times in 1986. Neither of these albums fared well on the charts or in terms of sales, with only modest success for the singles that were released from them (another duet with Dara, a cover of "Your Precious Love", reached the adult contemporary charts); Sedaka left Curb in 1986.

Other successes
American singer-songwriter Ben Folds credited Sedaka on his iTunes Originals album as an inspiration for his own song-publishing career. When Folds heard that Sedaka had a song published by the age of 13, Folds set a similar goal, despite the fact that Sedaka did not actually publish until he was 16.

In 1985, songs composed by Sedaka were adapted for the Japanese anime television series Mobile Suit Zeta Gundam. These included the two opening themes "Zeta-Toki wo Koete" (based on Sedaka's "Better Days Are Coming") and "Mizu no Hoshi e Ai wo Komete" (originally in English as "For Us to Decide", but the English version was never recorded), as well as the end theme "Hoshizora no Believe" (based on Sedaka's "Bad and Beautiful"). Due to copyright restrictions, the songs were replaced for the North American DVD, as well as for Japanese online releases of the series until 2017.

In 1994, Sedaka provided the voice for Neil Moussaka, a parody of himself in Food Rocks, an attraction at Epcot from 1994 to 2006.

A musical comedy based on the songs of Sedaka, Breaking Up Is Hard to Do, was written in 2005 by Erik Jackson and Ben H. Winters; it is now under license to Theatrical Rights Worldwide.

A biographical musical, Laughter in the Rain, produced by Bill Kenwright and Laurie Mansfield and starring Wayne Smith as Sedaka, had its world premiere at the Churchill Theatre in the London borough of Bromley on March 4, 2010. Sedaka attended the opening and joined the cast onstage for an impromptu curtain call of the title song.

Into the 21st century
Sedaka was inducted into the Songwriters Hall of Fame in 1983, has a star on the Hollywood Walk of Fame, and was an October 2006 inductee of the Long Island Music Hall of Fame. On November 15, 2013, Pacific Pioneer Broadcasters in Los Angeles gave him their Art Gilmore Career Achievement Award at a luncheon in his honor.

American Idol
In May 2003, near the end of the second season of the Fox TV series American Idol, Sedaka appeared as a guest judge and mentor to the five remaining finalists. Third-place finalist Kimberley Locke sang the Sedaka/Greenfield composition "Theme from Where the Boys Are".

Season 2 runner-up Clay Aiken chose Sedaka's 1972 song "Solitaire" for his performance. As Aiken explained to the studio and TV audiences, host Ryan Seacrest, and the four regular judges, "Solitaire" had long been one of his mother's all-time favorite songs. When she learned that Sedaka was going to be a guest judge and that the finalists would be singing Sedaka's songs, she begged Clay to sing "Solitaire". The performance was uniformly given extraordinarily high praise by the judges. Sedaka told Aiken that he officially passed ownership of the performance of "Solitaire" to Clay, offering to record and produce a single of the song or an entire CD with him.

Although it did not appear on his debut CD itself, Aiken recorded and added "Solitaire" as the B-side to the single "The Way", whose sales were faltering. "Solitaire" was quickly moved to the A-side, and radio airplay and single and download sales responded immediately. "Solitaire" hit No. 1 on the Billboard Hot Singles Sales chart and was, in fact, the top-selling single for all of 2004. It also hit the Top 5 on Billboard's Hot 100. Sedaka was invited back to American Idol to celebrate the success of "Solitaire" several times, as it continued to reach new milestones. Since then, Aiken has mined the Sedaka songbook again, recording a cover of probably Sedaka's best-known song, "Breaking Up Is Hard to Do", on the "deluxe version" of his 2010 CD release, Tried and True.

Amarillo – Guinness World Record
On a business trip to New York in mid-1971, Harvey Lisberg, who was a longtime fan of Sedaka, asked Don Kirshner if he'd written anything new. Kirshner took Lisberg to a small room with a piano where Sedaka was already seated, and he tapped out a few songs. One of these was the Sedaka/Greenfield composition "(Is This the Way to) Amarillo?" which Lisberg loved and placed with his artist Tony Christie who recorded and released it in 1971. The song did relatively well on the UK Singles Chart, reaching the Top 20.

Sedaka released the song in the U.S. in 1977 as the shortened "Amarillo", but it was only a mid-chart entry, peaking just shy of the Top 40.

When interviewed for an "extras" feature for a DVD set of a concert filmed in London on April 7, 2006 (see below), Sedaka jokingly said that he had heard that Christie had retired and was "golfing in Spain." The sudden revival of "Amarillo" summoned Christie back to the UK for an unexpected return to fame. In early 2006, the song received new life yet again when a dance beat was added and the lyrics were revised to become a novelty hit, released as "Is This the Way to the (England) World Cup?", to mark the appearance of the England football team at that summer's FIFA World Cup finals.

On April 7, 2006, Sedaka was appearing at the Royal Albert Hall and filming for the aforementioned CD/DVD package, when he was interrupted mid-concert by a gentleman who walked onstage from the wings. The planned scenario was that Sedaka was to begin performing "Amarillo", and after one verse, the audience was to be surprised by the appearance of Christie for an eventual duet. At the interruption, a confused Sedaka asked, "What is this?" The interloper was a representative from Guinness World Records, and he was there to present Sedaka with an award from Guinness World Records: British Hit Singles and Albums for composing "(Is This the Way to) Amarillo?", the most successful UK single of the 21st century (up to that date). After the presentation, Sedaka proceeded into "Amarillo", Christie entered onstage to an eruption of cheers from the audience, and after the successful duet performance, the two men walked offstage together, triumphantly arm in arm, as the first half of Sedaka's concert came to a close.

New recording contract, new chart success
Since Sedaka had lost his recording contract in the mid-1980s, he had used his own business, Neil Sedaka Music, to finance the recording, production, and distribution of new CDs and repackaging of his existing catalog of music. Because of ongoing disputes with RCA Records over the ownership of Sedaka's original late 1950s/early 1960s hits, in 1991, Sedaka re-recorded those early recordings.

In early 2007, Sedaka signed his first recording contract in nearly two decades with Razor and Tie Records, a small-but-growing, New York-based independent label with a talent roster that also includes Joan Baez, Vanessa Carlton, Foreigner, Joe Jackson, and Ladysmith Black Mambazo. The first release was The Definitive Collection, a life-spanning compilation of his hits, along with outtakes and songs previously released but unavailable in CD or download format. It debuted in the Top 25 on Billboard's Top 200 Albums chart in May 2007, one of the highest-charting albums of his entire career. It also includes "It Hurts to Be in Love", Sedaka's version of the Gene Pitney hit.

Waking Up Is Hard to Do was Sedaka's next release with Razor and Tie, hitting the albums chart in May 2009. The CD was a children's album that used the melodies of many of Sedaka's best-known songs but changed the lyrics to fit the everyday lives of babies and toddlers, along with their parents, grandparents, babysitters, and other caregivers. The CD title is an example. Lastly, The Music of My Life entered the albums chart in February 2010 and comprised almost all new material. "Right or Wrong", co-written with original music partner Howard Greenfield, was done in traditional street-corner, layered doo-wop vocal harmonies with Sedaka overlaying his own voice to achieve the effect for which he was well known in his "early" heyday of the late 1950s and early 1960s. The final track, "You", has been previously released, but was remastered for this project and is one of several titles dedicated to his wife and career guide of over 50 years, Leba.

A concert performance on October 26, 2007, at the Lincoln Center in New York City paid homage to the 50th anniversary of Sedaka's debut in show business. Music impresario (and producer for The Music of My Life track "Do You Remember?") David Foster served as emcee. Other guests included The Captain and Tennille; Natalie Cole; Connie Francis; recording legend and decades-long Sedaka friend and former manager Don Kirshner; and new Solitaire "owner" Clay Aiken, among many others. Also in 2007, Donny Osmond released a CD, Love Songs of the '70s, which included a cover of Sedaka's 1975 No. 1 hit "Laughter in the Rain".

During his 2008 Australian tour, Sedaka premiered a new classical orchestral composition entitled "Joie de Vivre" (Joy of Life). Sedaka also toured the Philippines for his May 17, 2008, concert at the Araneta Coliseum.

On September 11, 2010, Sedaka performed to a TV audience at the Hyde Park, London, venue of the "Proms in the Park" for the BBC.

In early 2011, Sedaka recorded two duets ("Brighton" and "The Immigrant") with singer Jim Van Slyke for Van Slyke's Neil Sedaka tribute album, The Sedaka Sessions. LML Records released this album in August 2011.

In 2010, Sedaka duetted with West End (London) and Broadway theatre legend Elaine Paige, on their cover of "Make It With You", from Ms. Paige's UK CD release Elaine Paige and Friends. The track was originally a No. 1 hit on the Billboard Hot 100 pop charts for the soft-rock group Bread in 1970. In 2014, Sedaka duetted with Engelbert Humperdinck. The pair recorded the title song from Sedaka's 1975 album The Hungry Years. The track is from Engelbert's UK/USA CD release Engelbert Calling. Also in 2014, Sedaka duetted with up-and-coming Nashville star Mary Sarah (Gross) on Connie Francis' classic, "Theme from Where the Boys Are", on her all-duets CD Bridges. Mary Sarah was also a Season 10 (spring 2016) contestant on NBC's The Voice, and once again relied on Sedaka's "Where the Boys Are" for her blind audition. She turned all four chairs around, and ultimately placed seventh in the competition. In 2015, Sedaka duetted on his No. 1 hit from 1974 to 1975, "Laughter in the Rain", with Steve Tyrell, on Tyrell's 2015 album That Lovin' Feeling. In 2014, BBC Four broadcast the documentary, Neil Sedaka: King of Song, telling the story of the two distinct periods of success in his career.

On February 1, 2016, Sedaka performed to a sold-out audience in The Villages. Then, on August 12, 2016, Sedaka released his new acoustic album, I Do It for Applause, which includes 11 new tracks and a bonus of his first symphony that was debuted in Australia in 2008, "Joie de Vivre (Joy of Life)"; the recording features the London Philharmonia Orchestra.

In April 2020, Sedaka launched a series of free mini-concerts, released through his social media channels, as a method of entertaining his fans during the COVID-19 pandemic. Each daily concert features three songs from Sedaka's discography. Sedaka paused the series in December due to contracting COVID-19 himself, but resumed on a reduced schedule January 4, 2021, after recovering with no symptoms.

Personal life
Sedaka attended Abraham Lincoln High School in Brooklyn, from which he graduated at the age of 17 in 1956.

He married Leba Strassberg in 1962. The couple have a daughter (Dara) and a son (Marc). Dara is a recording artist and vocalist for television and radio commercials who sang the female part on the Sedaka Billboard Top 20 hit duet, "Should've Never Let You Go" from 1980, and "Angel Queen" on the Queen Millennia soundtrack. Marc is a Los Angeles–based screenwriter who has three children with his wife, Samantha.

Sedaka's nephew, through his marriage to Strassberg, is CNN Politics writer Harry Enten.

Sedaka underwent a procedure to remove a benign skin tumor from his nose in March 2021.

As a performer

 * "Oh! Carol" (Italy, Netherlands and Wallonia), 1960
 * "One-Way Ticket to the Blues" (Japan), 1960
 * "Calendar Girl" (Canada & Japan), 1961
 * "Breaking Up Is Hard to Do" (1962 version) – (Canada), US Billboard Hot 100, 1962, Grammy nomination for Best R&R performance
 * "La terza luna" (Italy), 1963
 * "Star Crossed Lovers" (Australia), 1969
 * "Laughter in the Rain" – US Adult Contemporary, 1974; US Billboard Hot 100, 1975
 * "The Immigrant" – US Adult Contemporary, 1975
 * "Bad Blood" – US Billboard Hot 100, 1975–76, Grammy nomination for Best Pop Vocal Performance
 * "Breaking Up Is Hard to Do" (1975 version) – US Adult Contemporary, 1976, Grammy Award nomination for Song of the Year

As a songwriter

 * "Ring Ring" (English version) by ABBA (debut single) (songwriter) – (Belgium), 1973
 * "Solitaire" by The Carpenters (songwriter) – US Billboard Adult Contemporary, 1975
 * "Lonely Night (Angel Face)" by Captain & Tennille (1975) (songwriter) – (Canada), US Billboard Easy Listening
 * "Love Will Keep Us Together" by Captain & Tennille (1975) (songwriter) – US Billboard Hot 100, Grammy Award nomination for Song of the Year

Filmography

 * 1968: Playgirl Killer
 * 1973: The Sonny & Cher Comedy Hour (one episode, as himself) (musical performance)
 * 1976: Saturday Night Live (one episode, as himself) (musical performance)
 * 1977: The Carol Burnett Show (one episode, as himself)
 * 1980: The Toni Tennille Show (one episode, as himself)
 * 2005: The King of Queens (one episode, as himself)
 * 2020–present: Today's Mini-Concert (self-produced musical performance series)

Autobiography

 * Laughter in the Rain: My Own Story. New York: Putnam, 1982. ISBN 0-399-12744-5

title: Neil Sedaka


text:
Locally convex topological vector space

In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

Fréchet spaces are locally convex spaces that are completely metrizable (with a choice of complete metric). They are generalizations of Banach spaces, which are complete vector spaces with respect to a metric generated by a norm.

History
Metrizable topologies on vector spaces have been studied since their introduction in Maurice Fréchet's 1902 PhD thesis Sur quelques points du calcul fonctionnel (wherein the notion of a metric was first introduced). After the notion of a general topological space was defined by Felix Hausdorff in 1914, although locally convex topologies were implicitly used by some mathematicians, up to 1934 only John von Neumann would seem to have explicitly defined the weak topology on Hilbert spaces and strong operator topology on operators on Hilbert spaces. Finally, in 1935 von Neumann introduced the general definition of a locally convex space (called a convex space by him).

A notable example of a result which had to wait for the development and dissemination of general locally convex spaces (amongst other notions and results, like nets, the product topology and Tychonoff's theorem) to be proven in its full generality, is the Banach–Alaoglu theorem which Stefan Banach first established in 1932 by an elementary diagonal argument for the case of separable normed spaces (in which case the unit ball of the dual is metrizable).

Definition
Suppose $$X$$ is a vector space over $$\mathbb{K},$$ a subfield of the complex numbers (normally $$\Complex$$ itself or $\R$). A locally convex space is defined either in terms of convex sets, or equivalently in terms of seminorms.

Definition via convex sets
A topological vector space (TVS) is called  if it has a neighborhood basis (that is, a local base) at the origin consisting of balanced, convex sets. The term is sometimes shortened to  or.

A subset $$C$$ in $$X$$ is called
 * 1) Convex if for all $$x, y \in C,$$ and $$0 \leq t \leq 1,$$ $$tx + (1-t)y \in C.$$ In other words, $$C$$ contains all line segments between points in $$C.$$
 * 2) Circled if for all $$x \in C$$ and scalars $$s,$$ if $$|s| = 1$$ then $$s x \in C.$$  If $$\mathbb{K} = \R,$$ this means that $$C$$ is equal to its reflection through the origin.  For $$\mathbb{K} = \Complex,$$ it means for any $$x \in C,$$ $$C$$ contains the circle through $$x,$$ centred on the origin, in the one-dimensional complex subspace generated by $$x.$$
 * 3) Balanced if for all $$x \in C$$ and scalars $$s,$$ if $$|s| \leq 1$$ then $$s x \in C.$$  If $$\mathbb{K} = \R,$$ this means that if $$x \in C,$$ then $$C$$ contains the line segment between $$x$$ and $$-x.$$ For $$\mathbb{K} = \Complex,$$ it means for any $$x \in C,$$ $$C$$ contains the disk with $$x$$ on its boundary, centred on the origin, in the one-dimensional complex subspace generated by $$x.$$ Equivalently, a balanced set is a circled cone (in the TVS $ \mathbb R^2$, $C={}$ ball centered at the origin of radius $\sqrt2$ , $ x=(1,1)$  belongs, $s=2$ , $ sx=(2,2)$  does not belong, C is not a cone but C is balanced, sx is in C, for all x belonging to C and scalar s for which $|s|<1$ ).
 * 4) A cone (when the underlying field is ordered) if for all $$x \in C$$ and $$t \geq 0 ,$$ $$t x \in C.$$
 * 5) Absorbent or absorbing if for every $$x \in X,$$ there exists $$r > 0$$ such that $$x \in t C$$ for all $$t \in \mathbb{K}$$ satisfying $$|t| > r.$$ The set $$C$$ can be scaled out by any "large" value to absorb every point in the space.
 * 6) * In any TVS, every neighborhood of the origin is absorbent.
 * 7) Absolutely convex or a  if it is both balanced and convex. This is equivalent to it being closed under linear combinations whose coefficients absolutely sum to $$\leq 1$$; such a set is absorbent if it spans all of $$X.$$

In fact, every locally convex TVS has a neighborhood basis of the origin consisting of sets (that is, disks), where this neighborhood basis can further be chosen to also consist entirely of open sets or entirely of closed sets. Every TVS has a neighborhood basis at the origin consisting of balanced sets, but only a locally convex TVS has a neighborhood basis at the origin consisting of sets that are both balanced convex. It is possible for a TVS to have neighborhoods of the origin that are convex and yet not be locally convex because it has no neighborhood basis at the origin consisting entirely of convex sets (that is, every neighborhood basis at the origin contains some non-convex set); for example, every non-locally convex TVS $$X$$ has itself (that is, $$X$$) has a convex neighborhood of the origin.

Because translation is continuous (by definition of topological vector space), all translations are homeomorphisms, so every base for the neighborhoods of the origin can be translated to a base for the neighborhoods of any given vector.

Definition via seminorms
A seminorm on $$X$$ is a map $$p : X \to \R$$ such that

 * 1) $$p$$ is nonnegative or positive semidefinite: $$p(x) \geq 0$$;
 * 2) $$p$$ is positive homogeneous or positive scalable: $$p(s x) = |s| p(x)$$ for every scalar $$s.$$ So, in particular, $$p(0) = 0$$;
 * 3) $$p$$ is subadditive.  It satisfies the triangle inequality: $$p(x + y) \leq p(x) + p(y).$$

If $$p$$ satisfies positive definiteness, which states that if $$p(x) = 0$$ then $$x = 0,$$ then $$p$$ is a norm. While in general seminorms need not be norms, there is an analogue of this criterion for families of seminorms, separatedness, defined below.

If $$X$$ is a vector space and $$\mathcal{P}$$ is a family of seminorms on $$X$$ then a subset $$\mathcal{Q}$$ of $$\mathcal{P}$$ is called a base of seminorms for $$\mathcal{P}$$ if for all $$p \in \mathcal{P}$$ there exists a $$q \in \mathcal{Q}$$ and a real $$r > 0$$ such that $$p \leq r q.$$

Definition (second version): A locally convex space is defined to be a vector space $$X$$ along with a family $$\mathcal{P}$$ of seminorms on $$X.$$

Seminorm topology
Suppose that $$X$$ is a vector space over $$\mathbb{K},$$ where $$\mathbb{K}$$ is either the real or complex numbers. A family of seminorms $$\mathcal{P}$$ on the vector space $$X$$ induces a canonical vector space topology on $$X$$, called the initial topology induced by the seminorms, making it into a topological vector space (TVS). By definition, it is the coarsest topology on $$X$$ for which all maps in $$\mathcal{P}$$ are continuous.

It is possible for a locally convex topology on a space $$X$$ to be induced by a family of norms but for $$X$$ to be normable (that is, to have its topology be induced by a single norm).

Basis and subbases
Let $$B_{ 0$$ in $$\mathbb{K}$$. The family of sets $$p^{-1}\left(B_{<r}\right) = \{x \in X : p(x) < r\}$$ as $$p$$ ranges over a family of seminorms $$\mathcal{P}$$ and $$r$$ ranges over the positive real numbers is a subbasis at the origin for the topology induced by $$\mathcal{P}$$. These sets are convex, as follows from properties 2 and 3 of seminorms. Intersections of finitely many such sets are then also convex, and since the collection of all such finite intersections is a basis at the origin it follows that the topology is locally convex in the sense of the definition given above.

Recall that the topology of a TVS is translation invariant, meaning that if $$S$$ is any subset of $$X$$ containing the origin then for any $$x \in X,$$ $$S$$ is a neighborhood of the origin if and only if $$x + S$$ is a neighborhood of $$x$$; thus it suffices to define the topology at the origin. A base of neighborhoods of $$y$$ for this topology is obtained in the following way: for every finite subset $$F$$ of $$\mathcal{P}$$ and every $$r > 0,$$ let $$U_{F, r}(y) := \{x \in X : p(x-y) < r \ \text{ for all } p \in F\}.$$

Bases of seminorms and saturated families
If $$X$$ is a locally convex space and if $$\mathcal{P}$$ is a collection of continuous seminorms on $$X$$, then $$\mathcal{P}$$ is called a base of continuous seminorms if it is a base of seminorms for the collection of continuous seminorms on $$X$$. Explicitly, this means that for all continuous seminorms $$p$$ on $$X$$, there exists a $$q \in \mathcal{P}$$ and a real $$r > 0$$ such that $$p \leq r q.$$ If $$\mathcal{P}$$ is a base of continuous seminorms for a locally convex TVS $$X$$ then the family of all sets of the form $$\{x \in X : q(x) < r\}$$ as $$q$$ varies over $$\mathcal{P}$$ and $$r$$ varies over the positive real numbers, is a of neighborhoods of the origin in $$X$$ (not just a subbasis, so there is no need to take finite intersections of such sets).

A family $$\mathcal{P}$$ of seminorms on a vector space $$X$$ is called saturated if for any $$p$$ and $$q$$ in $$\mathcal{P},$$ the seminorm defined by $$x \mapsto \max \{ p(x), q(x) \}$$ belongs to $$\mathcal{P}.$$

If $$\mathcal{P}$$ is a saturated family of continuous seminorms that induces the topology on $$X$$ then the collection of all sets of the form $$\{x \in X : p(x) < r\}$$ as $$p$$ ranges over $$\mathcal{P}$$ and $$r$$ ranges over all positive real numbers, forms a neighborhood basis at the origin consisting of convex open sets; This forms a basis at the origin rather than merely a subbasis so that in particular, there is need to take finite intersections of such sets.

Basis of norms
The following theorem implies that if $$X$$ is a locally convex space then the topology of $$X$$ can be a defined by a family of continuous   on $$X$$ (a norm is a seminorm $$s$$ where $$s(x)=0$$ implies $$x=0$$) if and only if there exists  continuous    on $$X$$. This is because the sum of a norm and a seminorm is a norm so if a locally convex space is defined by some family $$\mathcal{P}$$ of seminorms (each of which is necessarily continuous) then the family $$\mathcal{P} + n := \{p + n : p \in \mathcal{P}\}$$ of (also continuous) norms obtained by adding some given continuous norm $$n$$ to each element, will necessarily be a family of norms that defines this same locally convex topology. If there exists a continuous norm on a topological vector space $$X$$ then $$X$$ is necessarily Hausdorff but the converse is not in general true (not even for locally convex spaces or Fréchet spaces).

$$

Nets
Suppose that the topology of a locally convex space $$X$$ is induced by a family $$\mathcal{P}$$ of continuous seminorms on $$X$$. If $$x \in X$$ and if $$x_{\bull} = \left(x_i\right)_{i \in I}$$ is a net in $$X$$, then $$x_{\bull} \to x$$ in $$X$$ if and only if for all $$p \in \mathcal{P},$$ $$p\left(x_{\bull} - x\right) = \left(p\left(x_i\right) - x\right)_{i \in I} \to 0.$$ Moreover, if $$x_{\bull}$$ is Cauchy in $$X$$, then so is $$p\left(x_{\bull}\right) = \left(p\left(x_i\right)\right)_{i \in I}$$ for every $$p \in \mathcal{P}.$$

Equivalence of definitions
Although the definition in terms of a neighborhood base gives a better geometric picture, the definition in terms of seminorms is easier to work with in practice. The equivalence of the two definitions follows from a construction known as the Minkowski functional or Minkowski gauge. The key feature of seminorms which ensures the convexity of their $$\varepsilon$$-balls is the triangle inequality.

For an absorbing set $$C$$ such that if $$x \in C,$$ then $$t x \in C$$ whenever $$0 \leq t \leq 1,$$ define the Minkowski functional of $$C$$ to be $$\mu_C(x) = \inf \{r > 0: x \in r C\}.$$

From this definition it follows that $$\mu_C$$ is a seminorm if $$C$$ is balanced and convex (it is also absorbent by assumption). Conversely, given a family of seminorms, the sets $$\left\{x : p_{\alpha_1}(x) < \varepsilon_1, \ldots, p_{\alpha_n}(x) < \varepsilon_n\right\}$$ form a base of convex absorbent balanced sets.

Ways of defining a locally convex topology
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Example: auxiliary normed spaces

If $$W$$ is convex and absorbing in $$X$$ then the symmetric set $$D := \bigcap_{|u|=1} u W$$ will be convex and balanced (also known as an or a ) in addition to being absorbing in $$X.$$ This guarantees that the Minkowski functional $$p_D : X \to \R$$ of $$D$$ will be a seminorm on $$X,$$ thereby making $$\left(X, p_D\right)$$ into a seminormed space that carries its canonical pseudometrizable topology. The set of scalar multiples $$r D$$ as $$r$$ ranges over $$\left\{\tfrac{1}{2}, \tfrac{1}{3}, \tfrac{1}{4}, \ldots\right\}$$ (or over any other set of non-zero scalars having $$0$$ as a limit point) forms a neighborhood basis of absorbing disks at the origin for this locally convex topology. If $$X$$ is a topological vector space and if this convex absorbing subset $$W$$ is also a bounded subset of $$X,$$ then the absorbing disk $$D := \bigcap_{|u|=1} u W$$ will also be bounded, in which case $$p_D$$ will be a norm and $$\left(X, p_D\right)$$ will form what is known as an auxiliary normed space. If this normed space is a Banach space then $$D$$ is called a.

Further definitions

 * A family of seminorms $$\left(p_{\alpha}\right)_{\alpha}$$ is called total or separated or is said to separate points if whenever $$p_{\alpha}(x) = 0$$ holds for every $$\alpha$$ then $$x$$ is necessarily $$0.$$ A locally convex space is Hausdorff if and only if it has a separated family of seminorms.  Many authors take the Hausdorff criterion in the definition.
 * A pseudometric is a generalization of a metric which does not satisfy the condition that $$d(x, y) = 0$$ only when $$x = y.$$ A locally convex space is pseudometrizable, meaning that its topology arises from a pseudometric, if and only if it has a countable family of seminorms.  Indeed, a pseudometric inducing the same topology is then given by $$d(x,y)=\sum^\infty_n \frac{1}{2^n} \frac{p_n(x-y)}{1+p_n(x-y)}$$ (where the $$1/2^n$$ can be replaced by any positive summable sequence $$a_n$$). This pseudometric is translation-invariant, but not homogeneous, meaning $$d(k x, k y) \neq |k| d(x, y),$$ and therefore does not define a (pseudo)norm. The pseudometric is an honest metric if and only if the family of seminorms is separated, since this is the case if and only if the space is Hausdorff. If furthermore the space is complete, the space is called a Fréchet space.
 * As with any topological vector space, a locally convex space is also a uniform space. Thus one may speak of uniform continuity, uniform convergence, and Cauchy sequences.
 * A Cauchy net in a locally convex space is a net $$\left(x_a\right)_{a \in A}$$ such that for every $$r > 0$$ and every seminorm $$p_\alpha,$$ there exists some index $$c \in A$$ such that for all indices $$a, b \geq c,$$ $$p_\alpha\left(x_a - x_b\right) < r.$$  In other words, the net must be Cauchy in all the seminorms simultaneously.  The definition of completeness is given here in terms of nets instead of the more familiar sequences because unlike Fréchet spaces which are metrizable, general spaces may be defined by an uncountable family of pseudometrics.  Sequences, which are countable by definition, cannot suffice to characterize convergence in such spaces.  A locally convex space is complete if and only if every Cauchy net converges.
 * A family of seminorms becomes a preordered set under the relation $$p_\alpha \leq p_\beta$$ if and only if there exists an $$M > 0$$ such that for all $$x,$$ $$p_\alpha(x) \leq M p_\beta(x).$$ One says it is a directed family of seminorms if the family is a directed set with addition as the join, in other words if for every $$\alpha$$ and $$\beta,$$ there is a $$\gamma$$ such that $$p_\alpha + p_\beta \leq p_\gamma.$$ Every family of seminorms has an equivalent directed family, meaning one which defines the same topology.  Indeed, given a family $$\left(p_\alpha(x)\right)_{\alpha \in I},$$ let $$\Phi$$ be the set of finite subsets of $$I$$ and then for every $$F \in \Phi$$ define $$q_F = \sum_{\alpha \in F} p_{\alpha}.$$ One may check that $$\left(q_F\right)_{F \in \Phi}$$ is an equivalent directed family.
 * If the topology of the space is induced from a single seminorm, then the space is seminormable. Any locally convex space with a finite family of seminorms is seminormable. Moreover, if the space is Hausdorff (the family is separated), then the space is normable, with norm given by the sum of the seminorms. In terms of the open sets, a locally convex topological vector space is seminormable if and only if the origin has a bounded neighborhood.

Hahn–Banach extension property
Let $$X$$ be a TVS. Say that a vector subspace $$M$$ of $$X$$ has the extension property if any continuous linear functional on $$M$$ can be extended to a continuous linear functional on $$X$$. Say that $$X$$ has the Hahn-Banach extension property (HBEP) if every vector subspace of $$X$$ has the extension property.

The Hahn-Banach theorem guarantees that every Hausdorff locally convex space has the HBEP. For complete metrizable TVSs there is a converse:

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If a vector space $$X$$ has uncountable dimension and if we endow it with the finest vector topology then this is a TVS with the HBEP that is neither locally convex or metrizable.

Properties
Throughout, $$\mathcal{P}$$ is a family of continuous seminorms that generate the topology of $$X.$$

Topological closure

If $$S \subseteq X$$ and $$x \in X,$$ then $$x \in \operatorname{cl} S$$ if and only if for every $$r > 0$$ and every finite collection $$p_1, \ldots, p_n \in \mathcal{P}$$ there exists some $$s \in S$$ such that $$\sum_{i=1}^n p_i(x - s) < r.$$ The closure of $$\{0\}$$ in $$X$$ is equal to $$\bigcap_{p \in \mathcal{P}} p^{-1}(0).$$

Topology of Hausdorff locally convex spaces

Every Hausdorff locally convex space is homeomorphic to a vector subspace of a product of Banach spaces. The Anderson–Kadec theorem states that every infinite–dimensional separable Fréchet space is homeomorphic to the product space $\prod_{i \in \N} \R$ of countably many copies of $$\R$$ (this homeomorphism need not be a linear map).

Properties of convex subsets
Algebraic properties of convex subsets

A subset $$C$$ is convex if and only if $$t C + (1 - t) C \subseteq C$$ for all $$0 \leq t \leq 1$$ or equivalently, if and only if $$(s + t) C = s C + t C$$ for all positive real $$s > 0 \text{ and } t > 0,$$ where because $$(s + t) C \subseteq s C + t C$$ always holds, the equals sign $$\,=\,$$ can be replaced with $$\,\supseteq.\,$$ If $$C$$ is a convex set that contains the origin then $$C$$ is star shaped at the origin and for all non-negative real $$s \geq 0 \text{ and } t \geq 0,$$ $$(s C) \cap (t C) = (\min_{} \{s, t\}) C.$$

The Minkowski sum of two convex sets is convex; furthermore, the scalar multiple of a convex set is again convex.

Topological properties of convex subsets

 * Suppose that $$Y$$ is a TVS (not necessarily locally convex or Hausdorff) over the real or complex numbers. Then the open convex subsets of $$Y$$ are exactly those that are of the form $$z + \{y \in Y : p(y) < 1\} = \{y \in Y : p(y - z) < 1\}$$ for some $$z \in Y$$ and some positive continuous sublinear functional $$p$$ on $$Y.$$
 * The interior and closure of a convex subset of a TVS is again convex.
 * If $$C$$ is a convex set with non-empty interior, then the closure of $$C$$ is equal to the closure of the interior of $$C$$; furthermore, the interior of $$C$$ is equal to the interior of the closure of $$C.$$
 * So if the interior of a convex set $$C$$ is non-empty then $$C$$ is a closed (respectively, open) set if and only if it is a regular closed (respectively, regular open) set.
 * If $$C$$ is convex and $$0 < t \leq 1,$$ then $$t \operatorname{Int} C + (1 - t) \operatorname{cl} C ~\subseteq~ \operatorname{Int} C.$$ Explicitly, this means that if $$C$$ is a convex subset of a TVS $$X$$ (not necessarily Hausdorff or locally convex), $$y$$ belongs to the closure of $$C,$$ and $$x$$ belongs to the interior of $$C,$$ then the open line segment joining $$x$$ and $$y$$ belongs to the interior of $$C;$$ that is, $$\{t x + (1 - t) y : 0 < t < 1\} \subseteq \operatorname{int}_X C.$$
 * If $$M$$ is a closed vector subspace of a (not necessarily Hausdorff) locally convex space$$X,$$ $$V$$ is a convex neighborhood of the origin in $$M,$$ and if $$z \in X$$ is a vector in $$V,$$ then there exists a convex neighborhood $$U$$ of the origin in $$X$$ such that $$V = U \cap M$$ and $$z \not\in U.$$
 * The closure of a convex subset of a locally convex Hausdorff space $$X$$ is the same for locally convex Hausdorff TVS topologies on $$X$$ that are compatible with duality between $$X$$ and its continuous dual space.
 * In a locally convex space, the convex hull and the disked hull of a totally bounded set is totally bounded.
 * In a complete locally convex space, the convex hull and the disked hull of a compact set are both compact.
 * More generally, if $$K$$ is a compact subset of a locally convex space, then the convex hull $$\operatorname{co} K$$ (respectively, the disked hull $$\operatorname{cobal} K$$) is compact if and only if it is complete.
 * In a locally convex space, convex hulls of bounded sets are bounded. This is not true for TVSs in general.
 * In a Fréchet space, the closed convex hull of a compact set is compact.
 * In a locally convex space, any linear combination of totally bounded sets is totally bounded.

Properties of convex hulls
For any subset $$S$$ of a TVS $$X,$$ the convex hull (respectively, closed convex hull, balanced hull, convex balanced hull) of $$S,$$ denoted by $$\operatorname{co} S$$ (respectively, $$\overline{\operatorname{co}} S,$$ $$\operatorname{bal} S,$$ $$\operatorname{cobal} S$$), is the smallest convex (respectively, closed convex, balanced, convex balanced) subset of $$X$$ containing $$S.$$

 * The convex hull of compact subset of a Hilbert space is necessarily closed and so also  necessarily compact. For example, let $$H$$ be the separable Hilbert space $\ell^2(\N)$ of square-summable sequences with the usual norm $$\|\cdot\|_2$$ and let $$e_n = (0, \ldots, 0, 1, 0, \ldots)$$ be the standard orthonormal basis (that is $$1$$ at the $$n^{\text{th}}$$-coordinate). The closed set $$S = \{0\} \cup \left\{\tfrac{1}{1} e_n, \tfrac{1}{2} e_2, \tfrac{1}{3} e_3, \ldots\right\}$$ is compact but its convex hull $$\operatorname{co} S$$ is  a closed set because $$h := \sum_{n=1}^{\infty} \tfrac{1}{2^n} \tfrac{1}{n} e_n$$ belongs to the closure of $$\operatorname{co} S$$ in $$H$$ but $$h \not\in\operatorname{co} S$$ (since every sequence $$z \in \operatorname{co} S$$ is a finite convex combination of elements of $$S$$ and so is necessarily $$0$$ in all but finitely many coordinates, which is not true of $$h$$). However, like in all complete Hausdorff locally convex spaces, the  convex hull $$K := \overline{\operatorname{co}} S$$ of this compact subset is compact. The vector subspace $$X := \operatorname{span} S$$ is a pre-Hilbert space when endowed with the substructure that the Hilbert space $$H$$ induces on it but $$X$$ is not complete and $$h \not\in C := K \cap X$$ (since $$h \not\in X$$). The closed convex hull of $$S$$ in $$X$$ (here, "closed" means with respect to $$X,$$ and not to $$H$$ as before) is equal to $$K \cap X,$$ which is not compact (because it is not a complete subset). This shows that in a Hausdorff locally convex space that is not complete, the closed convex hull of compact subset might  to be compact (although it will be precompact/totally bounded).
 * In a Hausdorff locally convex space $$X,$$ the closed convex hull $$\overline{\operatorname{co}}^X S = \operatorname{cl}_X \operatorname{co} S$$ of compact subset $$S$$ is not necessarily compact although it is a precompact (also called "totally bounded") subset, which means that its closure, $$\widehat{X}$$ of $$X,$$ will be compact (here $$X \subseteq \widehat{X},$$ so that $$X = \widehat{X}$$ if and only if $$X$$ is complete); that is to say, $$\operatorname{cl}_{\widehat{X}} \overline{\operatorname{co}}^X S$$ will be compact. So for example, the closed convex hull $$C := \overline{\operatorname{co}}^X S$$ of a compact subset of $$S$$ of a pre-Hilbert space $$X$$ is always a precompact subset of $$X,$$ and so the closure of $$C$$ in any Hilbert space $$H$$ containing $$X$$ (such as the Hausdorff completion of $$X$$ for instance) will be compact (this is the case in the previous example above).
 * In a quasi-complete locally convex TVS, the closure of the convex hull of a compact subset is again compact.
 * In a Hausdorff locally convex TVS, the convex hull of a precompact set is again precompact. Consequently, in a complete Hausdorff locally convex space, the closed convex hull of a compact subset is again compact.
 * In any TVS, the convex hull of a finite union of compact convex sets is compact (and convex).
 * This implies that in any Hausdorff TVS, the convex hull of a finite union of compact convex sets is (in addition to being compact and convex); in particular, the convex hull of such a union is equal to the  convex hull of that union.
 * In general, the closed convex hull of a compact set is not necessarily compact. However, every compact subset of $$\Reals^n$$ (where $$n < \infty$$) does have a compact convex hull.
 * In any non-Hausdorff TVS, there exist subsets that are compact (and thus complete) but closed.
 * The bipolar theorem states that the bipolar (that is, the polar of the polar) of a subset of a locally convex Hausdorff TVS is equal to the closed convex balanced hull of that set.
 * The balanced hull of a convex set is necessarily convex.
 * If $$C$$ and $$D$$ are convex subsets of a topological vector space $$X$$ and if $$x\in \operatorname{co} (C \cup D),$$ then there exist $$c \in C,$$ $$d \in D,$$ and a real number $$r$$ satisfying $$0 \leq r \leq 1$$ such that $$x = r c + (1 - r) d.$$
 * If $$M$$ is a vector subspace of a TVS $$X,$$ $$C$$ a convex subset of $$M,$$ and $$D$$ a convex subset of $$X$$ such that $$D \cap M \subseteq C,$$ then $$C = M \cap \operatorname{co} (C \cup D).$$
 * Recall that the smallest balanced subset of $$X$$ containing a set $$S$$ is called the balanced hull of $$S$$ and is denoted by $$\operatorname{bal} S.$$ For any subset $$S$$ of $$X,$$ the convex balanced hull of $$S,$$ denoted by $$\operatorname{cobal} S,$$ is the smallest subset of $$X$$ containing $$S$$ that is convex and balanced. The convex balanced hull of $$S$$ is equal to the convex hull of the balanced hull of $$S$$ (i.e. $$\operatorname{cobal} S = \operatorname{co} (\operatorname{bal} S)$$), but the convex balanced hull of $$S$$ is necessarily equal to the balanced hull of the convex hull of $$S$$ (that is, $$\operatorname{cobal} S$$ is not necessarily equal to $$\operatorname{bal} (\operatorname{co} S)$$).
 * If $$A, B \subseteq X$$ are subsets of a TVS $$X$$ and if $$s$$ is a scalar then $$\operatorname{co} (A + B) = \operatorname{co} (A) + \operatorname{co} (B),$$ $$\operatorname{co} (s A) = s \operatorname{co} A,$$ $$\operatorname{co} (A \cup B) = \operatorname{co} (A) \cup \operatorname{co} (B),$$ and $$\overline{\operatorname{co}}(s A) = s \overline{\operatorname{co}}(A).$$ Moreover, if $$\overline{\operatorname{co}}(A)$$ is compact then $$\overline{\operatorname{co}}(A + B) = \overline{\operatorname{co}}(A) + \overline{\operatorname{co}}(B).$$ However, the convex hull of a closed set need not be closed; for example, the set $$\left\{(x, \, \pm \tan x) : |x| < \tfrac{\pi}{2}\right\}$$ is closed in $$X := \R^2$$ but its convex hull is the open set $$\left(-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right) \times \R.$$
 * If $$A, B \subseteq X$$ are subsets of a TVS $$X$$ whose closed convex hulls are compact, then $$\overline{\operatorname{co}}(A \cup B) = \overline{\operatorname{co}}\left(\overline{\operatorname{co}}(A) \cup \overline{\operatorname{co}}(B)\right).$$
 * If $$S$$ is a convex set in a complex vector space $$X$$ and there exists some $$z \in X$$ such that $$z, iz, -z, -iz \in S,$$ then $$r z + s i z \in S$$ for all real $$r, s$$ such that $$|r| + |s| \leq 1.$$ In particular, $$a z \in S$$ for all scalars $$a$$ such that $$|a|^2 \leq \tfrac{1}{2}.$$
 * Carathéodory's theorem: If $$S$$ is subset of $$\Reals^n$$ (where $$n < \infty$$) then for every $$x \in \operatorname{co} S,$$ there exist a finite subset $$F \subseteq S$$ containing at most $$n + 1$$ points whose convex hull contains $$x$$ (that is, $$|F| \leq n + 1$$ and $$x \in \operatorname{co} F$$).

Coarsest vector topology
Any vector space $$X$$ endowed with the trivial topology (also called the indiscrete topology) is a locally convex TVS (and of course, it is the coarsest such topology). This topology is Hausdorff if and only $$X = \{0\}.$$ The indiscrete topology makes any vector space into a complete pseudometrizable locally convex TVS.

In contrast, the discrete topology forms a vector topology on $$X$$ if and only $$X = \{0\}.$$ This follows from the fact that every topological vector space is a connected space.

Finest locally convex topology
If $$X$$ is a real or complex vector space and if $$\mathcal{P}$$ is the set of all seminorms on $$X$$ then the locally convex TVS topology, denoted by $$\tau_{\operatorname{lc}},$$ that $$\mathcal{P}$$ induces on $$X$$ is called the  on $$X.$$ This topology may also be described as the TVS-topology on $$X$$ having as a neighborhood base at the origin the set of all absorbing disks in $$X.$$ Any locally convex TVS-topology on $$X$$ is necessarily a subset of $$\tau_{\operatorname{lc}}.$$ $$\left(X, \tau_{\operatorname{lc}}\right)$$ is Hausdorff. Every linear map from $$\left(X, \tau_{\operatorname{lc}}\right)$$ into another locally convex TVS is necessarily continuous. In particular, every linear functional on $$\left(X, \tau_{\operatorname{lc}}\right)$$ is continuous and every vector subspace of $$X$$ is closed in $$\left(X, \tau_{\operatorname{lc}}\right)$$; therefore, if $$X$$ is infinite dimensional then $$\left(X, \tau_{\operatorname{lc}}\right)$$ is not pseudometrizable (and thus not metrizable). Moreover, $$\tau_{\operatorname{lc}}$$ is the Hausdorff locally convex topology on $$X$$ with the property that any linear map from it into any Hausdorff locally convex space is continuous. The space $$\left(X, \tau_{\operatorname{lc}}\right)$$ is a bornological space.

Examples of locally convex spaces
Every normed space is a Hausdorff locally convex space, and much of the theory of locally convex spaces generalizes parts of the theory of normed spaces. The family of seminorms can be taken to be the single norm. Every Banach space is a complete Hausdorff locally convex space, in particular, the $L^p$ spaces with $$p \geq 1$$ are locally convex.

More generally, every Fréchet space is locally convex. A Fréchet space can be defined as a complete locally convex space with a separated countable family of seminorms.

The space $$\R^{\omega}$$ of real valued sequences with the family of seminorms given by $$p_i \left(\left\{x_n\right\}_n\right) = \left|x_i\right|, \qquad i \in \N$$ is locally convex. The countable family of seminorms is complete and separable, so this is a Fréchet space, which is not normable. This is also the limit topology of the spaces $$\R^n,$$ embedded in $$\R^{\omega}$$ in the natural way, by completing finite sequences with infinitely many $$0.$$

Given any vector space $$X$$ and a collection $$F$$ of linear functionals on it, $$X$$ can be made into a locally convex topological vector space by giving it the weakest topology making all linear functionals in $$F$$ continuous. This is known as the weak topology or the initial topology determined by $$F.$$ The collection $$F$$ may be the algebraic dual of $$X$$ or any other collection. The family of seminorms in this case is given by $$p_f(x) = |f(x)|$$ for all $$f$$ in $$F.$$

Spaces of differentiable functions give other non-normable examples. Consider the space of smooth functions $$f : \R^n \to \Complex$$ such that $$\sup_x \left|x^a D_b f\right| < \infty,$$ where $$a$$ and $$B$$ are multiindices. The family of seminorms defined by $$p_{a,b}(f) = \sup_x \left|x^a D_b f(x)\right|$$ is separated, and countable, and the space is complete, so this metrizable space is a Fréchet space. It is known as the Schwartz space, or the space of functions of rapid decrease, and its dual space is the space of tempered distributions.

An important function space in functional analysis is the space $$D(U)$$ of smooth functions with compact support in $$U \subseteq \R^n.$$ A more detailed construction is needed for the topology of this space because the space $$C_0^{\infty}(U)$$ is not complete in the uniform norm. The topology on $$D(U)$$ is defined as follows: for any fixed compact set $$K \subseteq U,$$ the space $$C_0^{\infty}(K)$$ of functions $$f \in C_0^{\infty}$$ with $$\operatorname{supp}(f) \subseteq K$$ is a Fréchet space with countable family of seminorms $$\|f\|_m = \sup_{k \leq m} \sup_x \left|D^k f(x)\right|$$ (these are actually norms, and the completion of the space $$C_0^{\infty}(K)$$ with the $$\| \cdot \|_m$$ norm is a Banach space $$D^m(K)$$). Given any collection $$\left(K_a\right)_{a\in A}$$ of compact sets, directed by inclusion and such that their union equal $$U,$$ the $$C_0^{\infty}\left(K_a\right)$$ form a direct system, and $$D(U)$$ is defined to be the limit of this system. Such a limit of Fréchet spaces is known as an LF space. More concretely, $$D(U)$$ is the union of all the $$C_0^{\infty}\left(K_a\right)$$ with the strongest topology which makes each inclusion map $$C_0^{\infty}\left(K_a\right) \hookrightarrow D(U)$$ continuous. This space is locally convex and complete. However, it is not metrizable, and so it is not a Fréchet space. The dual space of $$D\left(\R^n\right)$$ is the space of distributions on $$\R^n.$$

More abstractly, given a topological space $$X,$$ the space $$C(X)$$ of continuous (not necessarily bounded) functions on $$X$$ can be given the topology of uniform convergence on compact sets. This topology is defined by semi-norms $$\varphi_K(f) = \max \{|f(x)| : x \in K\}$$ (as $$K$$ varies over the directed set of all compact subsets of $$X$$). When $$X$$ is locally compact (for example, an open set in $$\R^n$$) the Stone–Weierstrass theorem applies—in the case of real-valued functions, any subalgebra of $$C(X)$$ that separates points and contains the constant functions (for example, the subalgebra of polynomials) is dense.

Examples of spaces lacking local convexity
Many topological vector spaces are locally convex. Examples of spaces that lack local convexity include the following:

 * The spaces $L^p([0, 1])$ for $$0 < p < 1$$ are equipped with the F-norm $$\|f\|^p_p = \int_0^1 |f(x)|^p \, dx.$$ They are not locally convex, since the only convex neighborhood of zero is the whole space. More generally the spaces $$L^p(\mu)$$ with an atomless, finite measure $$\mu$$ and $$0 < p < 1$$ are not locally convex.
 * The space of measurable functions on the unit interval $$[0, 1]$$ (where we identify two functions that are equal almost everywhere) has a vector-space topology defined by the translation-invariant metric (which induces the convergence in measure of measurable functions; for random variables, convergence in measure is convergence in probability): $$d(f, g) = \int_0^1 \frac{|f(x) - g(x)|}{1+|f(x) - g(x)|} \, dx.$$ This space is often denoted $$L_0.$$

Both examples have the property that any continuous linear map to the real numbers is $$0.$$ In particular, their dual space is trivial, that is, it contains only the zero functional.

 * The sequence space $$\ell^p(\N),$$ $$0 < p < 1,$$ is not locally convex.

Continuous mappings
$$

Because locally convex spaces are topological spaces as well as vector spaces, the natural functions to consider between two locally convex spaces are continuous linear maps. Using the seminorms, a necessary and sufficient criterion for the continuity of a linear map can be given that closely resembles the more familiar boundedness condition found for Banach spaces.

Given locally convex spaces $$X$$ and $$Y$$ with families of seminorms $$\left(p_\alpha\right)_{\alpha}$$ and $$\left(q_\beta\right)_{\beta}$$ respectively, a linear map $$T : X \to Y$$ is continuous if and only if for every $$\beta,$$ there exist $$\alpha_1, \ldots, \alpha_n$$ and $$M > 0$$ such that for all $$v \in X,$$ $$q_\beta(Tv) \leq M \left(p_{\alpha_1}(v) +\dotsb+p_{\alpha_n}(v)\right).$$

In other words, each seminorm of the range of $$T$$ is bounded above by some finite sum of seminorms in the domain. If the family $$\left(p_\alpha\right)_{\alpha}$$ is a directed family, and it can always be chosen to be directed as explained above, then the formula becomes even simpler and more familiar: $$q_\beta(Tv) \leq Mp_\alpha(v).$$

The class of all locally convex topological vector spaces forms a category with continuous linear maps as morphisms.

Linear functionals
$$

If $$X$$ is a real or complex vector space, $$f$$ is a linear functional on $$X$$, and $$p$$ is a seminorm on $$X$$, then $$|f| \leq p$$ if and only if $$f \leq p.$$ If $$f$$ is a non-0 linear functional on a real vector space $$X$$ and if $$p$$ is a seminorm on $$X$$, then $$f \leq p$$ if and only if $$f^{-1}(1) \cap \{x \in X : p(x) < 1\} = \varnothing.$$

Multilinear maps
Let $$n \geq 1$$ be an integer, $$X_1, \ldots, X_n$$ be TVSs (not necessarily locally convex), let $$Y$$ be a locally convex TVS whose topology is determined by a family $$\mathcal{Q}$$ of continuous seminorms, and let $$M : \prod_{i=1}^n X_i \to Y$$ be a multilinear operator that is linear in each of its $$n$$ coordinates. The following are equivalent:

 * 1) $$M$$ is continuous.
 * 2) For every $$q \in \mathcal{Q},$$ there exist continuous seminorms $$p_1, \ldots, p_n$$ on $$X_1, \ldots, X_n,$$ respectively, such that $$q(M(x)) \leq p_1\left(x_1\right) \cdots p_n\left(x_n\right)$$ for all $$x = \left(x_1, \ldots, x_n\right) \in \prod_{i=1}^n X_i.$$
 * 3) For every $$q \in \mathcal{Q},$$ there exists some neighborhood of the origin in $$\prod_{i=1}^{n} X_{i}$$ on which $$q \circ M$$ is bounded.

title: Locally convex topological vector space