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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 $(X,R,+,\xb7)$ 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 ${\alpha}_{i}=\u2329x,,,{e}_{i}\u232a$ .
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 ${c}_{k}=\u2329x,,,{e}_{k}\u232a=x(k/\left(2B\right))$ . 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 ${x}_{n}={\sum}_{i=1}^{n}\u2329x,,,{e}_{i}\u232a{e}_{i}$ 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: $\u2329x,,,y\u232a={\sum}_{i}\u2329x,,,{e}_{i}\u232a\overline{\u2329y,,,{e}_{i}\u232a}$ .
Using the CONS, we can write the partial sums ${x}_{n}={\sum}_{i=1}^{n}\u2329x,,,{e}_{i}\u232a{e}_{i}$ and ${y}_{n}={\sum}_{i=1}^{n}\u2329y,,,{e}_{i}\u232a{e}_{i}$ . We then have
Letting $n\to \infty $ we have that the upper bound goes to zero, and therefore as $n\to \infty $ , we have $\u2329{x}_{n},,,{y}_{n}\u232a\to \u2329x,,,y\u232a$ . Therefore,
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