# 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.

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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I don't know of any standard notation for such things, Andrew. $V\oplus V$ is often called the square of $V$, so for the second one I would just say that $V$ is isomorphic to its square. Being shiftable looks much stronger for separable Banach spaces than being isomorphic to hyperplanes. Is it really stronger? – Bill Johnson Nov 28 '11 at 22:56
Bill, I think that it is stronger because of the intersection property. If $X$ is isomorphic to a hyperplane in $X$, so $X \cong X \oplus \mathbb{R}$ then for any $Y$ we have the same property for $X \oplus Y$. But that's not true for a "shiftable space". If you don't know of a standard notation, then I'd take that as fairly definitive so I recommend that you post it as an answer. If someone comes along later and has an example of this in use then I can always edit the nLab page accordingly (it's a wiki, after all!). – Loop Space Nov 29 '11 at 7:48

## 1 Answer

I don't know of any standard notation for such things, Andrew. $V\oplus V$ is often called the square of $V$, so for the second one I would just say that $V$ is isomorphic to its square.

Being shiftable looks much stronger for separable Banach spaces than being isomorphic to hyperplanes. Is it really stronger? Is $\ell_p \oplus \ell_r$ shiftable when $p\not= r$?

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Regarding your last comment, on the nLab page I made a modification where I allowed for "shiftable of order k", so then l_p + l_r would be shiftable of order 2. For the applications I have in mind, I need "enough" spare space and too much is fine. – Loop Space Nov 29 '11 at 19:37