Suppose I have a Hilbert space equipped with the inner product $(x, y)$. If I introduce a new inner product $[x,y]$, can I say that this is equivalent to a certain change of basis, i.e., there is a $T$ such that $(x,y)=[Tx,Ty]$?
For finite-dimensional cases, we may just choose $\{\bf{e}_i\}$ to be an orthonormal basis with regard to the old inner product, and $\{\bf{f}_i\}$ to the new one. Then we simply define $T\bf{e}_i=\bf{f}_i$ to shuttle between the new and odd inner products.
What about the infinite-dimensional case?
