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"The primary attraction [of writing sf] is the sheer pleasure of creating something from whole cloth."
- Dan Simmons

  A realm or parallel universe in which it is possible to travel much faster than light.  

This is perhaps the first use of the term "hyperspace."

Are we not justified in supposing, by the analogy of the reversibility of reactions in chemistry, that the boundary lines of space and hyper-space may not be so rigidly drawn as we have supposed?

(The Invisible Bubble, by Kirk Meadowcroft)

Technovelgy from The Invisible Bubble, by Kirk Meadowcroft.
Published by Amazing Stories in 1928
Additional resources -

Here's another quotation:

We have discovered by my synthesis the factors necessary to produce a disturbance of spatial relation. We see that they may occur in nature; that under certain conditions, by the aid of lightning as well as by my tube, there may be produced the globules of hyper-space, which then seek each other out - flow together - and travel in vague and erratic fashion, as they may be drawn here and there by one influence or another.

Hyperspace as usually presented today is as an alternative to "normal space," that being the reality that we humans occupy.

John W. Campbell used it just three years later sans hyphen in Islands of Space:

Well we would be stuck if we didn't have this new method, for the acceleration would take too long. However, in the hyperspace we are going into, a new condition exists. Well you go into a hyperspace and move at any speed you please. I wonder then, how are you going to see where you are going?

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Additional resources:
  More Ideas and Technology from The Invisible Bubble
  More Ideas and Technology by Kirk Meadowcroft
  Tech news articles related to The Invisible Bubble
  Tech news articles related to works by Kirk Meadowcroft

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