Unitary Mapping

Unitary Mapping

A mapping $u:H_1\to H_2$ between two Hilbert spaces is called unitary if it is a vector space isomorphism and preserves the inner product, i.e. $⟨u(x),u(y){⟩}_{H_2}=⟨x,y{⟩}_{H_1}$for all $x,y\in H_1$. We call two Hilbert spaces unitarily equivalent if there exists a unitary mapping between them.

Theorem

Any two separable Hilbert spaces are unitarily equivalent if and only if they have the same dimension.

Proof

Proposition

$L^2(S^1)$ is unitarily equivalent to ${\ell }^2(\mathbb{Z})$ under the mapping $f\mapsto \{a_n(f)\}$, where $a_n(f)=\frac{1}{2\pi }{\int }_{\pi }^{\pi }f(x)e^{-inx}\text{dd}x$.

Proof