Every vector space over R or C can be made into an inner product space
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.