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**mysteriouspoet3000** Let V be a vector space over F = R or C.

For the case when dim V over F is finite, say n, if we let B be a basis for V,

a vector v in V has a corresponding coordinate vector with respect to the ordered basis B. Denote this representation as [v] = (a_1 ... a_n).

It is not difficult to check that

f(v,w) = sum of (conjugate(a_i) times b_i) where [v] = (a_1 ... a_n)

and [w]=(b_1 ... b_n) will work as an inner product.

Hence, V is an inner product space over F.

But I don't have an idea on what inner product to consider for the case when V is infinite dimensional over F= R or C.