Riesz representation theorem on Hilbert space with equivalent norms

Question

If we have a Hilbert space that has two equivalent norms (and inner products), are the Riesz maps (from Riesz representation theorem) associated with each inner product the same?

Answer 1

Note: of course, it fails already in $\mathbb{C}$. Just consider $(z,w)=z\overline{w}$ and $(z,w)'=2z\overline{w}$. And you see how to generalize this dilation idea. But I thought it could be more interesting to give the slightly more general viewpoint below.

If $(x,y)$ is one inner product and if $P$ is invertible in $B(H)$, then $$ (x,y)':=(Px,Py) $$ is an equivalent inner product. If $P$ is not an isometry for the latter, take $x$ such that $\|Px\|\neq \|x\|$. Then $$ (x,x)'=(Px,Px)=\|Px\|^2\neq \|x\|^2=(x,x). $$ In particular, the functionals $(x,\cdot )'$ and $(x,\cdot )$ are distinct.