Given a (separable) Banach space $(X,||.||)$, is there an inner product $<,>$ on $X$ such that $(X,||.||)$ and $(X,<,>)$ are linearly isomorphic?

Question

Let $(X,||.||)$ be a Banach space . From Can every Banach space be given an inner product homeomorphically? I know that there is an inner product on $X$ which generates the same topology as that of $(X,||.||)$ . My question is : does there always exist an inner product on $X$ giving it a norm $||.||_1$ such that $(X,||.||)$ and $(X,||.||_1)$ are linearly isomorphic i.e. there is a continuous, bijective , linear map between them with a continuous inverse ? If this is not always true , then what if we start with a separable Banach space $(X,||.||)$ ?

Answer 1

s.harp, in his/her comment, provides a correct answer. A more detailed explanation/reference for why $\ell^r$ is not linearly homeomorphic to $\ell^2$ for $2 < r$ can be found here Proof of Pitt's theorem.

A different approach would be to exhibit a non-reflexive separable Banach space. A Banach space is called reflexive if the natural embedding of $X$ in $X^{**}$ is surjective. This is true for Hilbert spaces and for $L^p$ spaces, for $1 < p < \infty$, and is a property preserved under linear homeomorphism. $c_0$ and $\ell_1$ are separable and non-reflexive, https://en.wikipedia.org/wiki/Reflexive_space#Examples

Edit: In response to a question, here is how to show that reflexivity is preserved under linear homeomorphism. Let $T: X \to Y$ be a linear homeomorphism of Banach spaces. Then $T^* : Y^* \to X^*$ and $T^{**} : Y^{**} \to X^{**}$ are linear homeomorphisms. Let $\iota_X$ and $\iota_Y$ be the natural embeddings of $X$ and $Y$ into their double dual spaces. Then check that $\iota_Y\circ T = T^{**} \circ \iota_X$; hence, $\iota_Y = T^{**} \circ \iota_X\circ T^{-1}$. Thus if $\iota_X$ is surjective, so is $\iota_Y$.