Can every Banach space be given an inner product homeomorphically?

Question

Let $(X,||.||)$ be a real Banach space , then is it true that there exists a norm $||.||_1$ on $X$ coming from an inner product such that $||.|| , ||.||_1$ generates same topology on $X$ or atleast $(X,||.||)$ and $(X,||.||_1)$ are homeomorphic ?

Answer 1

This is not the case. If it were, then $\mathrm{id}: (X,\|\cdot\|)\to (X,\|\cdot\|_1)$ would be a linear isomorphism and $(X,\|\cdot\|_1)$ a reflexive space, hence, as reflexivity is preserved by linear isomorphism, $(X,\|\cdot\|)$ would be a reflexive space, but this is not the case for all Banach spaces.

Answer 2

No, not every Banach space is isomorphic to a Hilbert space. Specific example is $c_0$.

As the OP is interested in non-linear homeomorphisms, then the answer is positive. See my answer here.

Answer 3

The following answers your last question. You can obtain this way a Hilbert space $(X,||.||_1)$ such that the spaces $(X,||.||)$ and $(X,||.||_1)$ are indeed homeomorphic (not by the identity map, of course). This follows

H. Toruńczyk, Characterizing Hilbert space topology. Fund. Math. 111 (1981), no. 3, 247–262.

In the separable setting this was proven earlier by M.I.Kadec:

"A proof of the topological equivalence of all separable infinite-dimensional Banach spaces", Funct. Anal. and Appl., Vol. 1 (1967).