I have to show that two Hilbert spaces are isomorphic using the definition professor gave us:
Two Hilbert spaces are isomorphic if there exists linear transformation $A: H_1 \rightarrow H_2$ such that $A$ is invertible and it preserves inner product.
I wanted to use this statement to show that seperable Hilbert space is isomorphic to $l^2$.
As linearity and preservation is not that hard, my main concern is about $A$ being invertible. How do I show that $A: H \rightarrow l^2$ defined below
$$Ax = (\langle x,e_k\rangle)_{k=1}^{\infty}$$
where $(e_k)$ is a basis of $H$, is indeed invertible?
Would be grateful for any hints.
Best Answer
Hints:
Let $(a_n)_n \in l^2$.
Finally, note that you showed that $A$ preserves the inproduct, so it preserves the distance too. Consequently, $A$ is an isometry and thus injective.