Best approximation theorem

Theorem

Let X be an inner product space with induced norm, and $Asubseteq X$ a non-empty, complete convex subset. Then, for all $xin X$ , there exists a unique best approximation a₀ to x in A.

Proof

Suppose x = 0 (if not the case, consider A − {x} instead) and let $d=d(0,A)=inf_{ain A} ||a||$ . There exists a sequence (a_n) in Asuch that

$||a_n||to d$ .

We now prove that (a_n) is a Cauchy sequence. By the parallelogram rule, we get

$||frac{a_n-a_m}{2}||^2+||frac{a_n+a_m}{2}||^2=frac{1}{2}(||a_n||^2+||a_m||^2)$ .

Since A is convex, $frac{a_n+a_m}{2}in A$ so

$underset{m,nin mathbb N}{forall }; ||frac{a_n+a_m}{2}||geq d$ .

Hence

$||frac{a_n-a_m}{2}||^2leq frac{1}{2}(||a_n||^2+||a_m||^2)-d^2to 0$ as $m,nto infty$

which implies $||a_n-a_m||to 0$ as $m,nto infty$ . In other words, (a_n) is a Cauchy sequence. Since A is complete,

$underset{a_0in A}{exists }; a_nto a_0$ .

Since $a_0in A$ , $||a_0||geq d$ . Furthermore

$||a_0||leq ||a_0-a_n||+||a_n||to d$ as $nto infty$ ,

which proves | | a₀ | | = d. Existence is thus proved. We now prove uniqueness. Suppose there were two distinct best approximations a₀and a₀‘ to x (which would imply | | a₀ | | = | | a₀‘ | | = d). By the parallelogram rule we would have

$||frac{a_0+a_0'}{2}||^2+||frac{a_0-a_0'}{2}||^2=frac{1}{2}(||a_0||^2+||a_0'||^2)=d^2$ .

Then

$||frac{a_0+a_0'}{2}||^2<d^2$

which cannot happen since A is convex, and as such $frac{a_0+a_0'}{2}in A$ , which means $||frac{a_0+a_0'}{2}||^2geq d^2$ , thus completing the proof.

Spectral theorem

From Wikipedia, the free encyclopedia

In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can bediagonalized (that is, represented as a diagonal matrix in some basis). This concept of diagonalization is relatively straightforward for operators on finite-dimensional spaces, but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of linear operators that can be modelled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical perspective.

Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operators on Hilbert spaces.

The spectral theorem also provides a canonical decomposition, called the spectral decomposition, eigenvalue decomposition, or eigendecomposition, of the underlying vector space on which the operator acts.

In this article we consider mainly the simplest kind of spectral theorem, that for a self-adjoint operator on a Hilbert space. However, as noted above, the spectral theorem also holds for normal operators on a Hilbert space.

http://en.wikipedia.org/wiki/Spectral_theorem