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"?