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Faster than Light travel (beating ultimate speed limit of nature) is common to Sci-Fi, especially Space Opera. Other names can be warping etc.

Which Sci-Fi work introduced this idea?

Note: It's possible that Greeks worked even on this. Religious texts can also mention it. But, such super-luminous travel barely has any significance in the context of the question because ultimate speed limit wasn't theorized until 1905 (when Einstein's Special Theory of Relativity released; although few decades old electromagnetism theories predicted the speed limit due to standing EM wave problem, it was far from acceptance). So, only post-1905 Sci-Fi works would be valid.

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    Do you count Lovecraft as "science fiction"? If so, then I believe the answer is "The Whisperer in Darkness", written in 1931 – Jason Baker Apr 3 '15 at 19:51
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    @JasonBaker Lovecraft wrote science fiction, but 1931 is much too late, E. E. Smith and Edmond Hamilton were writing FTL stories in the 1920s. – user14111 Apr 3 '15 at 20:01
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    Are we going by date of publication or date of writing? I'm thinking of The Skylark of Space published in 1928 but supposedly written 10 years earlier. I guess we go by publication date, because that's more verifiable, and anyway the idea is "introduced" when it's published. Anyway, I think Skylark is too late even if we go by the 1918-1919 date. – user14111 Apr 3 '15 at 21:10
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    Many of the early works of fiction made no real distinction of FTL vs STL drives. They just moved from point A to B in short periods of time. We now know that the velocity required FTL speeds to do the trip in the stated time. – Jim2B Apr 4 '15 at 1:21
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    Around a Distant Star by Jean Delaire. From Bleiler's review: "On returning from India, the narrator visits Roy and finds that he has been working on a spaceship and has devised a way to leave the earth. After studying the work of Tesla and Roentgen, and other areas of modern science, he discovered that a very powerful dynamo could generate positive electricity of sufficient force to toss a ship off the earth at two thousand times the speed of light. This renders interstellar travel possible." – user14111 Apr 4 '15 at 6:16
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Just before the 1905 cutoff date, in 1904, there was the novel Around a Distant Star by Jean Delaire. Quoting from Everett F. Bleiler's review in Science-Fiction: The Early Years:

On returning from India, the narrator visits Roy and finds that he has been working on a spaceship and has devised a way to leave the earth. After studying the work of Tesla and Roentgen, and other areas of modern science, he discovered that a very powerful dynamo could generate positive electricity of sufficient force to toss a ship off the earth at two thousand times the speed of light. This renders interstellar travel possible.

This seems to be the last of the pre-Einsteinian FTL stories. It's hard to find a good example of an FTL story for the next two decades. (I don't count stories where FTL is achieved by miraculous or supernatural means, or where it was all a dream.) The earliest post-Einsteinian examples I can find are Edmond Hamilton's "Crashing Suns" (first story in his Interstellar Patrol series), serialized in Weird Tales starting with the August 1928 issue (available at the Internet Archive), and Edward E. Smith's (with some assistance from Mrs. Lee Hawkins Garby) The Skylark of Space (first novel in Smith's Skylark series), serialized in Amazing Stories starting with the August 1928 issue (also available at the Internet Archive).

The Skylark of Space was significantly revised for book publication in 1946. The text of the original 1928 magazine version is available from Project Gutenberg. The following Wikipedia plot synopsis is based on the revised version:

The Skylark of Space is the first book of the Skylark series and pits the idealistic protagonist, Dick Seaton, against the mercantile antagonist Marc "Blackie" DuQuesne.

At the beginning of the story, Seaton accidentally discovers a workable space drive in combining pure copper with a newly discovered [fictional] element "X" (suggested to be a stable transactinide element in the platinum group) in solution. Having failed to re-create the effect, Seaton realizes that the missing component is a field generated by DuQuesne's particle accelerator, and thereafter sets up a business with his millionaire friend, Martin Crane, to build a spaceship. DuQuesne conspires to sabotage Seaton's spaceship and build his own from Seaton's plans, which he uses to kidnap Seaton's fiancée, Dorothy Vaneman, to exchange for the "X". In the resulting fight, DuQuesne's ship is accidentally set to full acceleration on an uncontrolled trajectory, until the copper 'power bar' is exhausted at a vast distance from Earth's solar system. Using an "Object Compass" that once locked on an object, always points toward that object, Seaton and Crane follow DuQuesne in their own spaceship (the eponymous Skylark) to rescue Dorothy and her fellow-hostage, Margaret "Peg" Spencer, until the Skylark discovers DuQuesne's ship derelict in orbit around a massive dead star (resembling a cold neutron star). Having obtained the hostages, Seaton extracts a promise from DuQuesne to "act as one of the party until they get back to Earth", in which relationship they leave orbit and travel further in search of additional fuel.

On an Earthlike exoplanet, they obtain "X" from an outcrop almost purely of that mineral; then leave that planet in search of copper. Following an encounter with a "Disembodied Intelligence" (Star Trek's "Q" would later show similar attributes), they enter a cluster of stars nicknamed “The Green System” and locate a planet having copper sulfate oceans. On the Earth-like "Osnome", they befriend the rulers of Mardonale, one of the two factions of the Osnomian natives. When the Mardonalian ruler attempts to betray Seaton and his friends, they find allies in Prince Dunark (a crown-prince of Mardonale's rival "Kondal") and his consort Princess Sitar, whom they later assist in destroying Mardonale. In gratitude, the Kondalians make new copper "power bars" and rebuild the Skylark as Skylark Two, with new weapons known to Kondalian science. Thereafter Seaton's marriage to Dorothy, and Crane's to Margaret, are solemnized by the Kondalian monarchy, and Seaton himself declared nominal "Overlord" of Kondal. The Skylark then returns to Earth, laden with jewels, radium, and a plenitude of "X"; but near Earth, DuQuesne leaves the Skylark by parachute, and the story ends with the Skylark's landing on Crane's Field.

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