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Let $V$ be an infinite dimensional vector space and $A$ be a subspace of $V$. Is there always an orthogonal complement of $A$ in $V$? If not, is there a counter-example? Thank you very much.

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What is an orthogonal complement? To have orthogonal complements, you need to have the notion of orthogonality of vectors, i.e. you need some kind of additional structure. An inner product, for example. Otherwise the notion is simply undefined. – Dan Shved Nov 19 '12 at 11:41
Perhaps there is a linear transformation you forgot to tell us about? – Gautam Shenoy Nov 19 '12 at 11:44
First of all, you need to fix a bilinear form $B$ to get a notion of orthogonality. If this is done, the set of vectors $x$ in $V$ such that $B(x,a)=0$ for all $a$ in $A$ is a subspace of $V$ and is defined to be the orthogonal complement of $A$ (w.r.t. $B$). I am confused about what your question is exactly. – Adeel Nov 19 '12 at 11:47
Maybe you are actually interested in complements as defined here, i.e. in the existence of a subspace $B$ such that $V=A \oplus B$? – Dan Shved Nov 19 '12 at 11:52
So you are interested in the existence of a (non degenerate) bilinear form on $V$? The set of bilinear forms forms a vector space which can be identified with the space of linear maps from $V$ to $V^*$; under this identification, non degenerate bilinear forms correspond to isomorphisms, so if $V \cong V^*$ then there is a nondegenerate form. If we don't require nondegenerateness then the map $A \mapsto A^\perp$ is not necessarily a bijection. – Adeel Nov 19 '12 at 12:53

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