Theorem of additional orthogonal of one closed in a space of Hilbert

The theorem of additional orthogonal of one closed in a space of Hilbert is a theorem of analyzes functional.

Statement

If $H$ is a Espace of Hilbert, and $F$ a vectorial subspace Fermé of $H$ , then the Orthogonal of $F$ is a additional Sous-espace of $F$ , i.e. $H = F \ oplus F^ \ bot$

Demonstration

$F$ is a vector space, therefore convex, and closed by assumption. One can thus apply the theorem of projection to convex closed:

$\ forall Z \ in H, \ exists! X \ in F, \|z-x \|= \ inf_ {U \ in F} \|z-u \|$

Let us note $y=z-x$ . One thus has:

\ forall U \ in F, \|z-u \|^2 = \|y+x-u \|^2 = \|there \|^2+ \|x-u \|^2+2 \ langle there, x-u \ rangle \ Ge \|there \|^2

from where $\ forall U \ in F, \|x-u \|^2 + 2 \ langle there, x-u \ rangle \ Ge 0$ .

Let us take $v$ such as $u=x+ \ epsilon v$ . The inequation becomes then

\ forall v \ in F, \ forall \ epsilon \ in R, \ epsilon^2 \|v \|^2+2 \ epsilon \ langle there, v \ rangle \ Ge 0

From where, by choosing a $\ epsilon$ initially then its opposite in the expression above:

$\ forall v \ in F, \ forall \ epsilon \ in R, \ begin {boxes} \ epsilon ^2 \|v \|^2+2 \ epsilon \ langle there, v \ rangle \ Ge 0 \ \ \ epsilon ^2 \|v \|^2-2 \ epsilon \ langle there, v \ rangle \ Ge 0 \ end {boxes}$

from where, while restricting themselves with $R^+$ and while simplifying by $\ epsilon$ :

\ forall v \ in F, \ forall \ epsilon \ in R^+, \ begin {boxes} \ epsilon \|v \|^2+2 \ langle there, v \ rangle \ Ge 0 \ \ \ epsilon \|v \|^2-2 \ langle there, v \ rangle \ Ge 0 \ end {boxes}

from where, while making tighten $\ epsilon$ towards 0,

\ forall v \ in F, \ langle there, v \ rangle =0

i.e.

y \ in F^ \ bot

And thus one can break up $z$ into

z=x+ there, X \ in F, there \ in F^ \ bot

As one always has $F \ course club-footed F^ \ = 0$ ,

one can thus finally conclude

H = F \ oplus F^ \ bot

Consequences

The $p_F$ application which has $z$ associated $x$ , above, defined by minimizing a distance, is called the projector on $F$ .

It has well the characteristic of a projector to the traditional direction (although definite topologically, and not algebraically), i.e. $p_F \ circ p_F=p_F$ .

When, as in our case, $H = F \ oplus F^ \ bot$ one calls this projector the orthogonal projector on $F$ , and one can specify core $F^ \ bot$ and of image $F$ .

One can then show that $p_F$ is linear .

Indeed, by unicity of the decomposition in two orthogonal and additional vectorial subspaces, if $z=x+ there, X \ in F, there \ in F^ \ bot$ , then $p_F (Z) =x$ .

Blow, are $z=x+y, z'=x'+y'$ the decompositions of two vectors, one from of deduced the decomposition $z+ \ lambda z'= (x+ \ lambda x') + (y+ \ lambda y')$ and thus $p_F (z+ \ lambda z') =x+ \ lambda x'=p_F (Z) + \ lambda p_F (z')$ .

One falls down of course the usual algebraic definition of a projector, which one knew well for spaces of finished size.

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