6.4 Hilbert spaces

Now we consider inner product spaces with nice convergence properties that allow us to define countably-infinite orthonormalbases.

• A Hilbert space is a complete inner product space. A complete
  The rational numbers provide an example of an incomplete set. We know thatit is possible to construct a sequence of rational numbers which approximate an irrational number arbitrarily closely. Itis easy to see that such a sequence will be Cauchy. However, the sequence will not converge to any rational number, and so the rationals cannot be complete.
  space is one where all Cauchy sequences converge to some vector within the space. For sequence $\{x_n\}$ to be Cauchy , the distance between its elements must eventually become arbitrarily small: $$\forall \epsilon , \epsilon > 0\colon \exists N_{\epsilon }\colon \forall n, m, (n\ge N_{\epsilon })\land (m\ge N_{\epsilon })\colon (x_n-x_m)< \epsilon$$ For a sequence $\{x_n\}$ to be convergent to x , the distance between its elements and $x$ must eventually become arbitrarily small: $$\forall \epsilon , \epsilon > 0\colon \exists N_{\epsilon }\colon \forall n, n\ge N_{\epsilon }\colon (x_n-x)< \epsilon$$ Examples are listed below (assuming the usual inner products):
  • $V=\mathbb{R}^N$
  • $V=\mathbb{C}^N$
  • $V=l_2$ ( i.e. , square summable sequences)
  • $V={ℒ}_2$ ( i.e. , square integrable functions)
• We will always deal with separable Hilbert spaces, which are those that have a countable
  A countable set is a set with at most a countably-infinite number of elements. Finite sets arecountable, as are any sets whose elements can be organized into an infinite list. Continuums ( e.g. , intervals of $\mathbb{R}$ ) are uncountably infinite.
  orthonormal (ON) basis. A countable orthonormal basis for $V$ is a countable orthonormal set $S=\{x_k\}$ such that every vector in $V$ can be represented as a linear combination of elements in $S$ : $$\forall y, y\in V\colon \exists \{{\alpha }_k\}\colon y=\sum_k {\alpha }_k{x}_k$$ Due to the orthonormality of $S$ , the basis coefficients are given by $${\alpha }_k=x_k\cdot y$$ We can see this via: $$x_k\cdot y=x_k\cdot \lim_{n\to }n\to$$∞ i 0 n α i x i n ∞ x k i 0 n α i x i n ∞ i 0 n α i x k x i α k where $\delta (k-i)=x_k\cdot x_i$ (where the second equality invokes the continuity of the inner product). In finite $n$ -dimensional spaces ( e.g. , $\mathbb{R}^n$ or $\mathbb{C}^n$ ), any $n$ -element ON set constitutes an ON basis. In infinite-dimensional spaces, wehave the following equivalences :
  • $\{x_0, x_1, x_2, \dots \}$ is an ON basis
  • If $x_i\cdot y=0$ for all $i$ , then $y=0$
  • $\forall y, y\in V\colon (y)^2=\sum_i \left|x_i\cdot y\right|^2$ (Parseval's theorem)
  • Every $y\in V$ is a limit of a sequence of vectors in $\mathrm{span}(\{x_0, x_1, x_2, \dots \})$
  Examples of countable ON bases for various Hilbert spaces include:
  • $\mathbb{R}^n$ : $\{e_0, \dots , e_{N-1}\}$ for $e_i=\begin{pmatrix}0 & \dots & 0 & 1 & 0 & \dots & 0\\ \end{pmatrix}^T$ with "1" in the $i^{\mathrm{th}}$ position
  • $\mathbb{C}^n$ : same as $\mathbb{R}^n$
  • $l_2$ : $\{\{{\delta }_i(n)\}\colon i\in \mathbb{Z}\}$ , for $≔(\{{\delta }_i(n)\}, \{\delta (n-i)\})$ (all shifts of the Kronecker sequence)
  • ${ℒ}_2$ : to be constructed using wavelets ...
• Say $S$ is a subspace of Hilbert space $V$ . The orthogonal complement of S in V , denoted $S^{\perp }$ , is the subspace defined by the set $\{x\in V\colon \perp (x, S)\}$ . When $S$ is closed, we can write $V=S\mathop{\mathrm{xor}}{S}^{\perp }$
• The orthogonal projection of y onto S , where $S$ is a closed subspace of $V$ , is $$\widehat{y}=\sum (x_i\cdot y)x_i$$ s.t. $\{x_i\}$ is an ON basis for $S$ . Orthogonal projection yields the best approximation of $y$ in $S$ : $$\widehat{y}=\mathrm{argmin}(x\in S, (y-x))$$ The approximation error $≔(e, y-\widehat{y})$ obeys the orthogonality principle : $$\perp (e, S)$$ We illustrate this concept using $V=\mathbb{R}^3$ ( ) but stress that the same geometrical interpretation applies to any Hilbertspace.

A proof of the orthogonality principle is: $$\iff (\perp (e, S), \forall i\colon e\cdot x_i=0)$$ $$y-\widehat{y}\cdot x_i=0$$