Not every norm on a vector space is induced by an inner product on that vector space. But suppose hat $X$ is a topological vector space that is normable, i.e. its topology is induced by some norm on the vector space. Then my question is, does that imply that $X$ is "inner product-able", i.e. that its topology is induced by some inner product on the vector space?
If not, then what is an example of a topological vector space which is normable but not "inner product-able"? And what properties must a topological vector space satisfy to be "inner product-able"?
Best Answer
$C[0,1]$ is normable with the sup norm. If the topology of $C[0,1]$ is induced by an inner product then the norm corresponding to the inner product is equivalent to the sup norm because these norms induce the same topology. But there is no norm equivalent to the sup norm which is given by an inner product. [ The existence of such a norm would make $C[0,1]$ reflexive].