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Does the same hold over an infinite dimensional vector space, provided that it has an algebraic basis? My insticts say no but I can't come up with a counterexample.

Here's one possible candidate. Let V be the set of all sequences [itex]\{x_i\}_{i\geq1}[/itex] over F for which only finitely many of the [itex]x_i[/itex] are nonzero. Let B be the following symmetric bilinear form:

[tex]B(\{x_i\},\{y_i\})=\sum_{i\geq1} (x_i y_{i+1}+ x_{i+1}y_i)[/tex]

which is a finite sum. Can you find a basis for V which is orthonormal wrt B? Can you prove that no such basis exists?

Thanks!