I been trying to figure out if the construction of the isomorphism between $g(W)$ and $f^{*}(W^{*})$ via $$B_V(v, g(w)) = B_W(f(v),w) \quad \forall v \in V, w \in W $$ described (at least this is what I think they mean) in the wikipedia page , is meaningfull in a Banach space which is not Hilbert. One reason it wouldn't be the lack of existancee of non-degenrate bilinear forms but this is something which I can't manage to establish. Or maybe there is some other reason?
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2$\begingroup$ I do not see any question in this question. $\endgroup$– gerwCommented Oct 25, 2015 at 18:31
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$\begingroup$ There is no need for the space to be Hilbert : you are adding this hypothesis from nowhere. Note: In the infinite case, an Hilbert space is not the same as a Banach space whose norm would come from a non degenerate product. $\endgroup$– MikeTeXCommented Oct 25, 2015 at 19:14
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$\begingroup$ The wiki article is about matrices. A continuous linear operator on a real Banach space may or may not be representable as a matrix $T$ .If it can, I don't know off-hand whether the transpose operator $T*(x)$ is definable at every $x$ in the space. There is a class called $B*$ algebras (and an important sub-class $C*$ algebras, which have "adjoints" which act like transposes in some ways,although bilinear forms (inner products) may not be available. This is a broad and deep subject. $\endgroup$– DanielWainfleetCommented Oct 25, 2015 at 19:35
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$\begingroup$ @MikeTex thats my point if the space is hilbert then evertying works out fine. But I think there might be Banach with non-degenerate biliear forms, and I wounder if there are adjonts in such spaces. $\endgroup$– user123124Commented Oct 27, 2015 at 8:12
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1$\begingroup$ @User2313: The concept you seem searching for goes under the name of Dual Topology for a Dual Pair. $\endgroup$– C-star-W-starCommented Nov 2, 2015 at 17:42
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