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I was wondering if linear manifold is a pure algebraic concept? Here is its definition from planetmath:

Suppose $V$ is a vector space and suppose that $L$ is a non-empty subset of $V$. If there exists a $v \in V$ such that $L+v=\{v+l, l \in L\}$ is a vector subspace of $V$ , then $L$ is a linear manifold of $V$.

If yes, why is there a "manifold" (which is a topological space) in its name?

Or is it a topological vector space with some topology unspecified but assumed

Thanks!

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Every finite-dimensional vector space$ V$ can be made into a manifold in a standard way, by choosing a vector space isomorphism between V and $\mathbb R^n$; the change-of- coordinate maps are linear and invertible. The condition on your $L$ is just so that it becomes a subspace by going thru the origin. –  gary Aug 16 '11 at 13:01
    
Linear manifold is a term used e.g. by Halmos and J.B. Conway to mean what is usually called linear subspace. In turn, they require a linear subspace to be closed (this is only of interest if you're working with infinite-dimensional topological vector spaces). I guess the reason is simply brevity: you need both concepts in functional analysis and always saying "closed subspace" or "not neccesarily closed subspace" is going to be mildly tedious. I don't think linear manifold is used that widely nowadays. –  t.b. Aug 16 '11 at 13:09
    
@Georges: I checked in Cristescu's book which planetmath cites as a source and in that book indeed an affine subspace is meant. Both Schaefer and Kelley-Namioka also use this terminology: from p.5 of Kelley-Namioka in the Springer GTM edition: "A set of the form $x + F$, where $F$ is a linear subspace, is called a linear manifold or linear variety, or a flat." I don't like to call affine things linear, but that may be my personal spleen. I don't know where the linear manifold terminology comes from. Banach himself had the good taste not to use it in his book. –  t.b. Aug 16 '11 at 13:50
    
Dear @Theo, thank you for these explanations. Like you, I don't like it when affine things are called linear. I had erased my comments before seeing your present answer because I didn't want to give the impression that I wanted a flame war against planetmath. Anyway, your comments nicely sum up what there is to say on the subject. –  Georges Elencwajg Aug 16 '11 at 14:22
    
@Georges: Don't think too much. Your comments are interesting and there will be no war. –  Tim Aug 16 '11 at 14:27

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