Theorem of continuity for linear mappings

Introduction

The theorem of continuity for linear mappings provides equivalent conditions for stiffness, with topology-producing functionals norms, seminorms, gauge functionals.

• Normed spaces - TCN The theorem of continuity for normed spaces is a special case of the more general case for topological vector spaces equivalent conditions are formulated for the stiffness of linear mappings over norms.
• Topological vector spaces - TCT This theorem generalizes the continuity of linear mapping for topological vector spaces and gauge functionals.

Linear mappings - finite dimensional vector spaces

A linear mapping ${\textstyle T:V\to W}$ of a finite dimensional vector space ${\textstyle V}$ over the field ${\textstyle \mathbb {K} }$ into a vector space ${\textstyle W}$ over the field ${\textstyle \mathbb {K} }$ is always continuous.

Linear mappings - not continuous

Linear mapping ${\textstyle T:V\to W}$ of an infinite-dimensional ${\textstyle \mathbb {K} }$-vector space ${\textstyle V}$ into a ${\textstyle \mathbb {K} }$ vector space ${\textstyle W}$ are also not continuous (see Examples).

Continuity for linear mapping - normed spaces

Let ${\textstyle (X,\|\cdot \|_{X})}$ and ${\textstyle (Y,\|\cdot \|_{Y})}$ normed spaces above the field ${\textstyle \mathbb {K} }$ and

$$T:X\rightarrow Y$$ a linear mapping, the following statements are equivalent:

• (1) T is steady at every point ${\textstyle x\in X}$
• (2) T is steady in the zero vector ${\textstyle 0_{X}\in X}$
• (3) There is a ${\textstyle M>0}$ with ${\textstyle \|T(x)\|_{Y}\leq M}$ for all ${\textstyle x\in X}$ with ${\textstyle \|x\|_{X}\leq 1}$
• (4) There is a ${\textstyle M>0}$ with ${\textstyle \|T(x)\|_{Y}\leq M\cdot \|x\|_{X}}$ for all ${\textstyle x\in X}$,

Proof

The proof of equivalence is performed by a cycle of implications in the following way (1) ${\textstyle \Rightarrow }$ (2) ${\textstyle \Rightarrow }$ (3) ${\textstyle \Rightarrow }$ (4) ${\textstyle \Rightarrow }$ (1)

Corrollary of TCN for bilinear mappings

The theorem of continuity can be transfered to bilinear mappings and normed spaces: Let ${\textstyle (X_{1},\|\cdot \|_{X_{1}})}$, ${\textstyle (X_{2},\|\cdot \|_{X_{2}})}$ and ${\textstyle (Y,\|\cdot \|_{Y})}$ normed spaces above the field ${\textstyle \mathbb {K} }$ and

$$T:X_{1}\times X_{2}\rightarrow Y$$ a bilinear mapping, the following statements are equivalent:

• (1) T is steady at every point ${\textstyle (x_{1},x_{2})\in X_{1}\times X_{2}}$
• (2) T is constantly in the zero vector ${\textstyle (0_{X_{1}},0_{X_{2}})\in X_{1}\times X_{2}}$
• (3) There is a ${\textstyle M>0}$ with ${\textstyle \|T(x_{1},x_{2})\|_{Y}\leq M}$ for all ${\textstyle (x_{1},x_{2})\in X}$ with ${\textstyle \|(x_{1},x_{2})\|:=\max\{\|x_{1}\|_{X_{1}},\|x_{2}\|_{X_{2}}\}\leq 1}$
• (4) There is a ${\textstyle M>0}$ with ${\textstyle \|T(x_{1},x_{2})\|_{Y}\leq M\cdot \|x_{1}\|_{X_{1}}\cdot \|x_{2}\|_{X_{2}}}$ for all ${\textstyle (x_{1},x_{2})\in X_{1}\times X_{2}}$,

Remark - Product space as vector space

The product space ${\textstyle X_{1}\times X_{2}}$ is naturally converted into a ${\textstyle \mathbb {K} }$-vector space through the following operations ${\textstyle \oplus ,\ \odot }$:

$${\begin{array}{rcl}(x_{1},x_{2})\oplus (y_{1},y_{2})&:=&(x_{1}+y_{1},x_{2}+y_{2})\\\lambda \odot (x_{1},x_{2})&:=&(\lambda \cdot x_{1},\lambda \cdot x_{2})\\\end{array}}$$

With ${\textstyle \|(x_{1},x_{2})\|:=\max\{\|x_{1}\|_{X_{1}},\|x_{2}\|_{X_{2}}\}}$ the product space ${\textstyle X_{1}\times X_{2}}$ also becomes a normed space.

Application of Corrolary

It is helpful for the Topologization lemma for algebras to prove the stiffness to a point. scalar multiplication and the multiplication on the algebra are in this context bilineare mappings. For example, ${\textstyle (X_{1},\|\cdot \|_{_{X_{1}}}):=(\mathbb {K} ,|\cdot |)}$ and ${\textstyle (X_{2},\|\cdot \|_{_{X_{2}}}):=(A,\|\cdot \|)}$ with ${\textstyle \|\cdot \|:A\to \mathbb {R} _{o}^{+}}$ are the submultiplicative standard on the algebra ${\textstyle A}$.

Task - Proof Corollary

Prove the above corollary using the ideas from the theorem of continuity for linear mappings on normed spaces. Notes:

• Use the equivalence of

${\textstyle {\begin{array}{rcl}\|(x_{1},x_{2})\|&:=&\max\{\|x_{1}\|_{X_{1}},\|x_{2}\|_{X_{2}}\}\leq 1\\&\Longleftrightarrow &\|x_{1}\|_{X_{1}}\leq 1\wedge \|x_{2}\|_{X_{2}}\leq 1\\\end{array}}}$.

• Use the linearity in each component to estimate ${\textstyle (3)\to (4)}$.

Task - equivalence of norms - product space

In the above corollar, a standard ${\textstyle \|(x_{1},x_{2})\|:=\max\{\|x_{1}\|_{X_{1}},\|x_{2}\|_{X_{2}}\}}$ is defined on ${\textstyle X_{1}\times X_{2}}$. Show that ${\textstyle \|(x_{1},x_{2})\|_{\ast }:=\|x_{1}\|_{X_{1}}+\|x_{2}\|_{X_{2}}}$ is a äquivalente Norm on ${\textstyle X_{1}\times X_{2}}$.

Operator standard

The condition (4) from the stiffness set for linear mappings leads to the introduction of the operator space. This makes the vector space of the steady linear functions ${\textstyle {\mathcal {L}}_{C}(V,W)}$ a subset of all linear mappings ${\textstyle {\mathcal {L}}(V,W)}$ itself a normedn space. (the index ${\textstyle C}$ in ${\textstyle {\mathcal {L}}_{C}(V,W)}$ stands for "continuous".

Alternative statement

Alternatively to (3), the statement can also be formulated as follows:

There is a ${\textstyle M>0}$ with ${\textstyle \|T\|_{\mathcal {L}}:=\sup\{\|T(x)\|_{Y}\,:\,\|x\|_{X}=1\}\leq M}$

This is equivalent to

698-1047-1747202468649-341-68

Definition: Operatornorm

Be ${\textstyle (V,\|\cdot \|_{V})}$ and ${\textstyle (W,\|\cdot \|_{W})}$ normed vector spaces above the field ${\textstyle \mathbb {K} }$ and ${\textstyle {\mathcal {L}}(V,W):=\{T\colon V\to W\mid T\ {\text{linear}}\}}$ the set of linear mapping of (698-1047-174720246 is ${\textstyle T\colon V\rightarrow W}$ linearer Operator. Then the operator standard

$$\|{\cdot }\|_{\mathcal {L}}\;\colon \;{\mathcal {L}}(V,W)\to \mathbb {R} _{0}^{+}\cup \{\infty \}$$

concerning Vektornormen ${\textstyle \|\cdot \|_{V}}$ and ${\textstyle \|\cdot \|_{W}}$ by

$$\|T\|_{\mathcal {L}}:=\inf \left\{M\geq 0\mid \forall v\in V\colon \|T(v)\|_{W}\leq M\,\|v\|_{V}\right\}$$

defined.

Comments - Operatornorm

The operator standard ${\textstyle \|T\|_{\mathcal {L}}}$ provides a smallest upper barrier for the stretching of vectors from the one-piece ball in ${\textstyle (X,\|\cdot \|_{X})}$.

Linear mappings with finite definition range

For finite-dimensional vector spaces, this distinction is not necessary because each finite-dimensional linear mapping is continuous.

Task 1

Prove the set that linear mappings with a finite definition range ${\textstyle V}$ are steady.

Evidence

Let${\textstyle dim(V)=n}$ and ${\textstyle B=(b_{1},\ldots ,b_{n})}$ have a base of nominated vectors for ${\textstyle V}$ (i.e. ${\textstyle \|b_{k}\|_{V}=1}$ for all ${\textstyle k\in \{1,\ldots ,n\}}$.

• Use the statement (3) from the grade for linear mappings.
• Select ${\textstyle v}$ from the completed single ball ${\textstyle {\overline {B_{1}^{\|\cdot \|_{V}}(0_{V})}}}$.
• Set ${\textstyle v}$ as Linearkombination of the base vectors.
• Estimate the standard ${\textstyle \left\|T(v)\right\|_{W}}$.

Note: Stability and Standard Limitation

For continuous linear mapping of a normaledn space ${\textstyle (V,\|\cdot \|_{V})}$ according to ${\textstyle (W,\|\cdot \|_{W})}$, the image ${\textstyle T\left({\overline {B_{1}^{\|\cdot \|_{V}}(0_{V})}}\right)}$ of the completed single ball ${\textstyle {\overline {B_{1}^{\|\cdot \|_{V}}(0_{V})}}}$ is limited to the standard (698-1047-174720246.

Stability set for linear mapping on topological vector spaces

Bee ${\textstyle (X,\|\cdot \|_{\mathcal {A}})}$ and ${\textstyle (Y,\|\cdot \|_{\widetilde {\mathcal {A}}})}$ topological vector spaces with the systems of topologieerzeugenden gauge functionals above the field ${\textstyle \mathbb {K} }$ and

$$T:X\rightarrow Y$$ a linear mapping, the following statements are equivalent:

• (1) T is steady at every point ${\textstyle x\in X}$
• (2) T is steady in the zero vector ${\textstyle 0_{X}\in X}$
• (3) ${\textstyle \forall _{{\widetilde {\alpha }}\in {\widetilde {\mathcal {A}}}}\exists _{\alpha \in {\mathcal {A}},\,M_{\alpha }>0}\forall _{x\in X}\,:\,\|x\|_{\alpha }\leq 1\Longrightarrow \|T(x)\|_{\widetilde {\alpha }}\leq M_{\alpha }}$
• (4) ${\textstyle \forall _{{\widetilde {\alpha }}\in {\widetilde {\mathcal {A}}}}\exists _{\alpha \in {\mathcal {A}},\,M_{\alpha }>0}\forall _{x\in X}\,:\,\|T(x)\|_{\widetilde {\alpha }}\leq M_{\alpha }\cdot \|x\|_{\alpha }}$,

Proof SLAT

Also the Stetigkeitssatz für Lineare mapping auf topologischen vector spaces (SLAT) becomes as Ringschluss of (1) ${\textstyle \Rightarrow }$ (2) ${\textstyle \Rightarrow }$ (3) ${\textstyle \Rightarrow }$ (4) ${\textstyle \Rightarrow }$ (1).

Korrollar SLAT for bilinear mappings

The assessment of the stiffness rate applies analogously to bilineare mappings and normed spaces: Bee ${\textstyle (X_{1},\|\cdot \|_{{\mathcal {A}}_{1}})}$, ${\textstyle (X_{2},\|\cdot \|_{{\mathcal {A}}_{2}})}$ and ${\textstyle (Y,\|\cdot \|_{\widetilde {\mathcal {A}}})}$ normed spaces above the field ${\textstyle \mathbb {K} }$ and

$$T:X_{1}\times X_{2}\rightarrow Y$$ a bilinear mapping, the following statements are equivalent:

• (1) T is steady at every point ${\textstyle (x_{1},x_{2})\in X_{1}\times X_{2}}$
• (2) T is steady in the zero vector ${\textstyle (0_{X_{1}},0_{X_{2}})\in X_{1}\times X_{2}}$

(3) ${\textstyle \forall _{{\widetilde {\alpha }}\in {\widetilde {\mathcal {A}}}}\exists _{\alpha _{1}\in {\mathcal {A}}_{1},\alpha _{2}\in {\mathcal {A}}_{2},\,M>0}\forall _{(x_{1},x_{2})\in X_{1}\times X_{2}}\,:\|T(x_{1},x_{2})\|_{\widetilde {\alpha }}\leq M}$ for all ${\textstyle (x_{1},x_{2})\in X_{1}\times X_{2}}$ with ${\textstyle \|(x_{1},x_{2})\|_{(\alpha _{1},\alpha _{1})}:=\max\{\|x_{1}\|_{\alpha _{1}},\|x_{2}\|_{\alpha _{2}}\}\leq 1}$

• (4) ${\textstyle \forall _{{\widetilde {\alpha }}\in {\widetilde {\mathcal {A}}}}\exists _{\alpha _{1}\in {\mathcal {A}}_{1},\alpha _{2}\in {\mathcal {A}}_{2},\,M>0}\forall _{(x_{1},x_{2})\in X_{1}\times X_{2}}\,:\,\|T(x_{1},x_{2})\|_{\widetilde {\alpha }}\leq M\cdot \|x_{1}\|_{\alpha _{1}}\cdot \|x_{2}\|_{\alpha _{2}}}$ for all ${\textstyle (x_{1},x_{2})\in X_{1}\times X_{2}}$,

measure functional and partial order

The index quantities ${\textstyle I}$ of the nets are selected as a function of the index quantity of the measured functionals. ${\textstyle I:={\mathcal {A}}\times \mathbb {R} ^{+}}$ is a suitable choice (see gauge functionale und partielle Ordnung.

See also

• Stetigkeitssatz für normed Räume
• Stetigkeitssatz für topological vector spaces
• bilineare mappings
• nets
• Hahn-Banach - normed Räume
• Functional Analysis
• Kontraposition
• Konvergenz in normedn Räumen
• normsäquivalenzsatz
• Stetigkeit in topologischen Räumen
• Submultiplikativität
• Measure Theory on topological spaces

Page Information

Translation and Version Control

This page was translated based on the following Wikiversity source page and uses the concept of Translation and Version Control for a transparent language fork in a Wikiversity:

• Source: Stetigkeitssatz für lineare Abbildungen
• URL: https://de.wikiversity.org/wiki/Stetigkeitssatz%20f%C3%BCr%20lineare%20Abbildungen
• Date: 5/14/2025 8:05