Showing that two Hilbert spaces are isomorphic

hilbert-spacesinner-productsvector-space-isomorphism

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 $(a_n)_n \in l^2$.

  • Show that $\sum_n a_n e_n$ converges in the Hilbert space.
  • Let $x:= \sum_n a_n e_n$.
  • Deduce that $a_n:= \langle x, e_n \rangle$
  • Conclude surjectivity.

Finally, note that you showed that $A$ preserves the inproduct, so it preserves the distance too. Consequently, $A$ is an isometry and thus injective.

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