I'm writing up some notes on the nLab about things like embedding spaces and infinite spheres and similar things (can't link to them yet as I haven't put them up yet). One aspect that crops up time and time again is the contractibility of some big space, such as an infinite sphere, and this almost always boils down to some special property of whatever topological vector space is sitting in the background.

This special property is the existence of a "shift map" which acts pretty much like the obvious shift map on a sequence space. So I'm going to refer often to pairs $(V,S)$ where $V$ is a locally convex topological vector space and $S \colon V \to V$ is a "shift map". A little more precisely, we want to have an isomorphism $V \cong V \oplus \mathbb{R}$, so that $S \colon V \to V$ is the inclusion of the first factor, with certain properties, the main one being that $\bigcap S^k V = {0}$. The obvious notation is that $(V,S)$ is a shift space, and that $V$ is a shiftable space, but if there's an already existent notation then I should use that.

So my question is that: is there a standard notation for any of these concepts? The map itself, the space that admits the map, and the pair.

A closely related concept that I'll also use a bit could be termed a split space. This would be a locally convex topological vector space $V$ with an isomorphism $V \cong V \oplus V$. So: same question for that.

Edit: As Bill Johnson hasn't heard of these, I've written the relevant pages. It may be that I've included some detail there that I didn't put here. If any further information comes to light, I'll edit the pages accordingly.

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# Is there a standard notation for a "shift space" in functional analysis?

I'm writing up some notes on the nLab about things like embedding spaces and infinite spheres and similar things (can't link to them yet as I haven't put them up yet). One aspect that crops up time and time again is the contractibility of some big space, such as an infinite sphere, and this almost always boils down to some special property of whatever topological vector space is sitting in the background.

This special property is the existence of a "shift map" which acts pretty much like the obvious shift map on a sequence space. So I'm going to refer often to pairs $(V,S)$ where $V$ is a locally convex topological vector space and $S \colon V \to V$ is a "shift map". A little more precisely, we want to have an isomorphism $V \cong V \oplus \mathbb{R}$, so that $S \colon V \to V$ is the inclusion of the first factor, with certain properties, the main one being that $\bigcap S^k V = {0}$. The obvious notation is that $(V,S)$ is a shift space, and that $V$ is a shiftable space, but if there's an already existent notation then I should use that.

So my question is that: is there a standard notation for any of these concepts? The map itself, the space that admits the map, and the pair.

A closely related concept that I'll also use a bit could be termed a split space. This would be a locally convex topological vector space $V$ with an isomorphism $V \cong V \oplus V$. So: same question for that.