1.13 Infinite-dimensional hilbert spaces

While up to now bases have been linked to finite-dimensional spaces and subspaces, it is possible to extend the concept to infinite-dimensional spaces.

Definition 1 Let $(X,R,+,·)$ be a vector space. An infinite sequence of orthonormal vectors $\{e_1,e_2,...\}\subseteq X$ is said to be a complete orthonormal sequence (CONS) in $X$ if for every $x\in X$ there exists a sequence ${\alpha }_1,{\alpha }_2,...\in R$ such that $x={\sum }_i{\alpha }_i{e}_i$ .

For the sake of concreteness, an infinite sum is defined as $x={\sum }_{i=1}^{\infty }{\alpha }_i{e}_i={lim}_{n\to \infty }{\sum }_{i=1}^n{\alpha }_i{e}_i$ . It is easy to see that ${\alpha }_i=〈x,,,e_i〉$ .

Lemma 1 An orthonormal sequence is complete if and only if the only vector in $X$ orthogonal to each of the $e_i$ 's is the zero vector.

Example 1 For the space $X$ of finite-energy complex-valued functions, $R=\mathbb{C}$ , a CONS is given by $e_k\left(t\right)={\displaystyle \frac{1}{\sqrt{2\pi }}}{e}^{jkt}$ for $k=0,\pm 1,\pm 2,...$ . These vectors are orthonormal:

The coefficients are given by

and we obtain $x={\sum }_k{c}_k{e}_k$ . This is the sequence behind the Fourier series representation.

Example 2 Let $X$ be the space of bandlimited functions $x\left(t\right)$ (i.e., the set of functions with Fourier transform $X\left(f\right)$ such that $\left|X\right(f\left)\right|=0$ for all $f\notin [-B,B]$ ). A CONS for this space is given by

where $\mathrm{sinc}\left(t\right)=(sin(\pi t\left)\right)/\left(\pi t\right)$ . It is possible to show that the functions are orthogonal to each other, i.e.,

If $x$ is bandlimited, then it follows that $x\left(t\right)={\sum }_k{c}_k{e}_k\left(t\right)$ , with $c_k=〈x,,,e_k〉=x(k/\left(2B\right))$ . The preservation of the norm in the coefficients can also be extended from ONBs to CONS.

Theorem 1 (Completeness Relation) An orthonormal sequence $e_1,e_2,...$ is complete for $X$ if and only if the completeness relation holds for all $x\in X$ :

The sequence $\left\{e_i\right\}$ is CONS if and only if

where $x_n={\sum }_{i=1}^n〈x,,,e_i〉e_i$ is the partial sum. We then have ${\parallel x\parallel }^2=\parallel x-x_n{\parallel }^2+{\parallel x_n\parallel }^2$ as these two components are orthogonal to each other. Applying a limit on both sides for $n$ , we have

Theorem 2 Let $X$ be a Hilbert space with a CONS $\{e_1,e_2,...\}$ . Then for any $x,y\in X$ , Parseval's relation holds: $〈x,,,y〉={\sum }_i〈x,,,e_i〉\overline{〈y,,,e_i〉}$ .

Using the CONS, we can write the partial sums $x_n={\sum }_{i=1}^n〈x,,,e_i〉e_i$ and $y_n={\sum }_{i=1}^n〈y,,,e_i〉e_i$ . We then have

Letting $n\to \infty$ we have that the upper bound goes to zero, and therefore as $n\to \infty$ , we have $〈x_n,,,y_n〉\to 〈x,,,y〉$ . Therefore,