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"I realized there was a heavy-duty change coming in biology, and I could write a really compelling novel about catastrophic evolution, punctuated equilibrium."
- Greg Bear

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

As far as I know, this is the first use of the term "hyperspace" in fiction.

AS a psychologist I was, and am, not unknown. My monographs on “The Will” and “The Fundamentals of Character” are accepted as authoritative. I have also some reputation as a mathematician, or rather that branch of it which is included in the domain of metaphysics, transcendental mathematics, which was developed by the mystics of antiquity and more recently by Zollner, Hinton, and Einstein.

Into the mysteries of hyperspace T have penetrated farther than any man now living. I am one of the few who can say that physically I have been in the fourth dimension. It was while in this plane that I learned that J. Ensley Kendall had betrayed me. I read his thoughts as if they were an open book. When I returned to three dimensional space I went to him and accused him of his treachery and upbraided him. He listened. It even seemed to me that he listened patiently.

Technovelgy from The Statement of Archibald Swayne, by Burton Peter Thom.
Published by Weird Tales in 1924
Additional resources -

Here's another quotation from The Invisible Bubble (1928), by Kirk Meadowcroft:

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)

...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 Statement of Archibald Swayne
  More Ideas and Technology by Burton Peter Thom
  Tech news articles related to The Statement of Archibald Swayne
  Tech news articles related to works by Burton Peter Thom

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