Strictly rationed space

The single ball on the middle figure is strictly convex, while the other two are not (their borders contain straight line segments).

In mathematics , strictly normalized spaces are an important subclass of normed spaces that are close in their structure to Hilbert spaces . For such spaces, the problem of uniqueness of approximations has been solved, and this property is widely used in the problems of computational mathematics and mathematical physics. In addition, in a strictly normalized space, the segment connecting two points of an arbitrary sphere will lie entirely inside (except for the boundary points) of the open ball bounded by the given sphere.

A normed space X is called strictly normalized (or strictly convex ) if for arbitrary $x,y\in X$ ${\ displaystyle x, y \ in X}$ $x, y \ in X$ satisfying the condition $\|x+y\|=\|x\|+\|y\|$ ${\ displaystyle \ | x + y \ | = \ | x \ | + \ | y \ |}$ ${\ displaystyle \ | x + y \ | = \ | x \ | + \ | y \ |}$ there is such $\lambda \in \mathbb {R}$ ${\ displaystyle \ lambda \ in \ mathbb {R}}$ ${\ displaystyle \ lambda \ in \ mathbb {R}}$ , what $y=\lambda x$ ${\ displaystyle y = \ lambda x}$ ${\ displaystyle y = \ lambda x}$ .

Properties of strictly normalized spaces

Let X be a strictly normed space, and L a linear subspace . Then for $\forall x\in X$ ${\ displaystyle \ forall x \ in X}$ there is no more than one element $u\in L$ ${\ displaystyle u \ in L}$ such that $\rho (x,L)=\|x-u\|$ ${\ displaystyle \ rho (x, L) = \ | xu \ |}$ .

Element $u$ ${\ displaystyle u}$ call the best approximation element x elements of L. The existence of the best approximation element is provided by the following theorem.

Theorem . Let X be a normed space , and L a finite-dimensional linear subspace. Then for $\forall x\in E$ ${\ displaystyle \ forall x \ in E}$ there is an element of best approximation $u\in L$ ${\ displaystyle u \ in L}$ .

At the same time, in a normalized, but not strictly normalized space, the element of best approximation, in general, is not unique.

Each ball of a strictly normalized space is a strictly convex set . The converse is also true; if in a normed space each ball is a strictly convex set, then this space is strictly normalized.
A normed space X is strictly normalized if and only if from the condition $\forall x,y\in X:\|x\|=1,\|y\|=1,x\neq y$ ${\ displaystyle \ forall x, y \ in X: \ | x \ | = 1, \ | y \ | = 1, x \ neq y}$ always follows that $\|x+y\|<2$ ${\ displaystyle \ | x + y \ | <2}$ .

Examples of strictly normalized spaces

$\mathbb {R} ^{2}$ ${\ displaystyle \ mathbb {R} ^ {2}}$ with the norm $\|\mathbf {x} \|_{2}={\sqrt {x_{1}^{2}+x_{2}^{2}}}$ ${\ displaystyle \ | \ mathbf {x} \ | _ {2} = {\ sqrt {x_ {1} ^ {2} + x_ {2} ^ {2}}}$ . However norms $\|\mathbf {x} \|_{1}=|x_{1}|+|x_{2}|$ ${\ displaystyle \ | \ mathbf {x} \ | _ {1} = | x_ {1} | + | x_ {2} |}$ and $\|\mathbf {x} \|_{\infty }=\max\{|x_{1}|,|x_{2}|\}$ ${\ displaystyle \ | \ mathbf {x} \ | _ {\ infty} = \ max \ {| x_ {1} |, | x_ {2} | \}}$ on $\mathbb {R} ^{2}$ ${\ displaystyle \ mathbb {R} ^ {2}}$ equivalent to the norm $\|\cdot \|_{2}$ ${\ displaystyle \ | \ cdot \ | _ {2}}$ do not generate strictly normalized space (see figure).
$L_{p}$ ${\ displaystyle L_ {p}}$ where ${\displaystyle 1<semantics><mrow class=$ ${\ displaystyle 1 <p <\ infty}$ . This fact follows from Young's inequality , which is used in deriving the Hölder and Minkowski inequalities .
Hilbert spaces

Literature

Trenogin V. А. Functional analysis. - M .: Science , 1980 . - 495 s.
Functional analysis / editor S.G. - 2nd, revised and enlarged. - M .: Science , 1972 . - 544 s. - (Mathematical Reference Library).

Strictly rationed space

Properties of strictly normalized spaces

Examples of strictly normalized spaces

Literature

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