Theorem of continuity for linear mappings

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.

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 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).

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

$${\displaystyle T:X\rightarrow Y}$$ 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 ${\textstyle x\in X}$,

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)

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

$${\displaystyle T:X_{1}\times X_{2}\rightarrow Y}$$ 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 ${\textstyle (x_{1},x_{2})\in X_{1}\times X_{2}}$,

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

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

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.

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 ${\textstyle A}$.

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 ${\textstyle (3)\to (4)}$.

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 ${\textstyle X_{1}\times X_{2}}$.

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".

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

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

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

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

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

defined.

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

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

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

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 ${\textstyle \left\|T(v)\right\|_{W}}$.

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.

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

$${\displaystyle T:X\rightarrow Y}$$ 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) ${\textstyle \forall _{{\widetilde {\alpha }}\in {\widetilde {\mathcal {A}}}}\exists _{\alpha \in {\mathcal {A}},\,M_{\alpha }>0}\forall _{x\in X}\,:\,\|T(x)\|_{\widetilde {\alpha }}\leq M_{\alpha }\cdot \|x\|_{\alpha }}$,

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).

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

$${\displaystyle T:X_{1}\times X_{2}\rightarrow Y}$$ 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 ${\textstyle (x_{1},x_{2})\in X_{1}\times X_{2}}$,

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.

• 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

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