Prof. Matthieu Bloch
Monday, September 9, 2024 (v1.1)
The following properties of a countable orthonormal set \(\set{e_i}_{i\geq 1}\) are equivalent:
A space is separable if it contains a countable dense subset.
Any separable Hilbert space has an orthonormal basis.
Any separable Hilbert space is isomorphic to \(\ell_2\); the isomorphism preserves the norm and inner product.
Let \(\set{v_i}_{i=1}^n\) be a linearly independent set in a Hilbert space \(\calH\) of dimension \(n\). Then, for any \(x\in\calH\), \(x=\sum_{i=1}^n\alpha_iv_i\) for some \(\bfalpha\in\bbR^n\). In addition, there exists \(A,B>0\) such that \[ A\norm[2]{\bfalpha}^2 \leq \norm[\calH]{x}^2\leq B\norm[2]{\bfalpha}^2 \]
For any \(x\in\calH\) with basis \(\set{v_i}_{i=1}^n\) we have \[ x=\sum_{i=1}^n\alpha_iv_i\qquad\text{with}\qquad\bfalpha = \matG^{-1}\mat{c}{\dotp{x}{v_1}\\\dotp{x}{v_2}\\\vdots\\\dotp{x}{v_n}} \] There also exists a basis \(\set{\tilde{v}_i}_{i=1}^n\) such that \(\alpha_i=\dotp{x}{\tilde{v}_i}\)
\(\set{v_i}_{i=1}^\infty\) is a Riesz basis for Hilbert space \(\calH\) if \(\text{cl}(\text{span}(\set{v_i}_{i=1}^\infty))=\calH\) and there exists \(A,B>0\) such that \[ A\sum_{i=1}^\infty\alpha_i^2\leq \norm[\calH]{\sum_{i=1}^n\alpha_iv_i}^2\leq B\sum_{i=1}^\infty\alpha_i^2 \] uniformly for all sequences \(\set{\alpha_i}_{i\geq 1}\) with \(\sum_{i\geq 1}\alpha_i^2<\infty\).
Computing expansion on Riesz basis not as simple in infinite dimension: Gram matrix is "infinite"
The Gramian is a linear operator \[ \calG:\ell_2(\bbZ)\to\ell_2(\bbZ): \bfx\mapsto \bfy\text{ with }[\calG(\bfx)]_n\eqdef y_n=\sum_{\ell=-\infty}^\infty\dotp{v_\ell}{v_n}x_\ell \]
Fact: there exists another linear operator \(\calH:\ell_2(\bbZ)\to\ell_2(\bbZ)\) such that \[ \calH(\calG(\bfx)) = \bfx \] We can replicate what we did in finite dimension!