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.

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 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".

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