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)
$V={\mathcal{L}}_{2}$ (
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:
$${x}_{k}\cdot y={x}_{k}\cdot \lim_{n\to}n\to $$∞i0nαixin∞xki0nαixin∞i0nαixkxiαk 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
$\u2254(\{{\delta}_{i}(n)\}, \{\delta (n-i)\})$ (all shifts of the Kronecker sequence)
${\mathcal{L}}_{2}$ : 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
$\u2254(e, y-\widehat{y})$ 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:
$$\iff (\perp (e, S), \forall i\colon e\cdot {x}_{i}=0)$$$$y-\widehat{y}\cdot {x}_{i}=0$$
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