Für die folgenden Abbildung
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{\displaystyle \|\!|\cdot |\!\|_{\alpha }:\to \mathbb {R} ^{+}}
Quasihalbnormen auf der Polynomalgebra
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{\displaystyle A[t]}
. Die Quasihalbnormeigenschaften und die Hausdorff-Eigenschaft auf
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{\displaystyle A[t]}
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{\displaystyle (A,\|\cdot \|_{\mathcal {A}})}
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{\displaystyle p\in A[t]}
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{\displaystyle \lambda \in \mathbb {K} }
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{\displaystyle {\begin{array}{rcl}\|\!|\lambda \cdot p|\!\|_{\alpha }&=&\displaystyle \sum _{n=0}^{\infty }C_{k}(\alpha )\cdot \left\|\lambda \cdot p_{k}\right\|_{\alpha }\\&=&\displaystyle |\lambda |\cdot \sum _{n=0}^{\infty }C_{k}(\alpha )\cdot \left\|p_{k}\right\|_{\alpha }\\&=&|\lambda |\cdot \|\!|\lambda \cdot p|\!\|_{\alpha }\end{array}}}
Gilt für
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{\displaystyle p\in A[t]}
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{\displaystyle p\not =0_{_{A[t]}}}
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{\displaystyle A[t]}
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{\displaystyle k\in \mathbb {N} _{0}}
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{\displaystyle p_{k}\not =0_{_{A}}}
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{\displaystyle (A,\|\cdot \|_{\mathcal {A}})}
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{\displaystyle \|\cdot \|_{\alpha }}
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{\displaystyle \alpha \in {\mathcal {A}}}
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{\displaystyle \|p_{k}\|_{\alpha }>0\Rightarrow C_{k}(\alpha )\cdot \|p_{k}\|_{\alpha }>0\Rightarrow \|\!|\lambda \cdot p|\!\|_{\alpha }>0}
Bearbeiten
Sei
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{\displaystyle K_{\alpha }\geq 1}
Quasihalbnorm .
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{\displaystyle {\begin{array}{rcl}\|\!|p+q|\!\|_{\alpha }&=&\displaystyle \sum _{n=0}^{\infty }C_{n}(\alpha )\cdot \left\|p_{k}+q_{k}\right\|_{\alpha }\\&\leq &\displaystyle \sum _{n=0}^{\infty }C_{n}(\alpha )\cdot K_{\alpha }\cdot \left(\left\|p_{k}\right\|_{\alpha }+\left\|q_{k}\right\|_{\alpha }\right)\\&=&\displaystyle K_{\alpha }\cdot \left(\sum _{n=0}^{\infty }C_{n}(\alpha )\cdot \left\|p_{n}\right\|_{\alpha }+\sum _{n=0}^{\infty }C_{n}(\alpha )\cdot \left\|q_{n}\right\|_{\alpha }\right)\\&=&K_{\alpha }\cdot \left(\|\!|p|\!\|_{\alpha }+\|\!|q|\!\|_{\alpha }\right)\end{array}}}
Mit der Stetigkeit der Multiplikation auf
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{\displaystyle (A,\|\cdot \|_{\mathcal {A}})}
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{\displaystyle \alpha \in {\mathcal {A}}}
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{\displaystyle \beta \in {\mathcal {A}}}
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{\displaystyle \|x\cdot y\|_{\alpha }\leq \|x\|_{\beta }\cdot \|y\|_{\beta }}
Diese Halbnorm wird nun verwendet, um auf für die Cauchy-Multiplkation auf
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{\displaystyle \|\!|\cdot |\!\|_{\beta }}
Koeffizientenlemma der Cauchy-Multiplikation
Bearbeiten
Mit dem Koeffizientenlemma der Cauchy-Multiplikation und
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{\displaystyle K_{\alpha }\geq 1}
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{\textstyle K\geq 1}
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{\displaystyle \|\cdot \|_{\alpha }}
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{\textstyle (C_{n}(\alpha ))_{n\in \mathbb {N} _{0}}}
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{\textstyle (D_{n}(\alpha ))_{n\in \mathbb {N} _{0}}}
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{\textstyle (C_{n}(\beta ))_{n\in \mathbb {N} _{0}}}
(KL1)
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{\textstyle C_{i}(\beta )\geq D_{i}(\alpha )}
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{\textstyle K_{\alpha }^{i+j+1}\cdot C_{i+j}(\alpha )\leq C_{i}(\beta )\cdot C_{j}(\beta )}
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{\textstyle i,j\in \mathbb {N} _{0}}
Stetigkeit der Multiplikation - Polynomalgebra
Bearbeiten
Mit der Stetigkeit der Multiplikation auf
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{\displaystyle A}
Koeffizientenlemmas der Cauchy-Multiplikation erhält man die Stetigkeit der Multiplikation auf
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{\displaystyle {\begin{array}{rcl}\|\!|p\cdot q|\!\|_{\alpha }&\leq &\displaystyle \sum _{n=0}^{\infty }C_{n}(\alpha )\cdot \left\|\sum _{k=0}^{n}p_{k}\cdot q_{n-k}\right\|_{\alpha }\\&\leq &\displaystyle \sum _{n=0}^{\infty }C_{n}(\alpha )\cdot K_{\alpha }^{n+1}\cdot \sum _{k=0}^{n}\left\|p_{k}\cdot q_{n-k}\right\|_{\alpha }\\&\leq &\displaystyle \sum _{n=0}^{\infty }\sum _{k=0}^{n}\underbrace {C_{n}(\alpha )\cdot K_{\alpha }^{n+1}} _{=C_{k}(\beta )\cdot C_{n-k}(\beta )}\cdot \left\|p_{k}\right\|_{\beta }\cdot \left\|q_{n-k}\right\|_{\beta }\\&=&\displaystyle \sum _{n=0}^{\infty }\sum _{k=0}^{n}C_{k}(\beta )\cdot \left\|p_{k}\right\|_{\beta }\cdot C_{n-k}(\beta )\cdot \left\|q_{n-k}\right\|_{\beta }\\&=&\|\!|p|\!\|_{\beta }\cdot \|\!|q|\!\|_{\beta }\end{array}}}
Mit der obigen Abschätzung für alles
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{\displaystyle \alpha \in {\mathcal {A}}}
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{\displaystyle (A[t],\|\!|\cdot |\!\|_{\mathcal {A}})}
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{\displaystyle \alpha ,\beta \in {\mathcal {A}}}
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{\displaystyle \left\|\cdot \right\|_{\mathcal {A}}}
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{\displaystyle \|\!|\cdot |\!\|_{\mathcal {A}}}
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{\displaystyle A[t]}
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{\displaystyle \alpha ,\beta \in {\mathcal {A}}}
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{\displaystyle (A[t],\|\!|\cdot |\!\|_{\mathcal {A}})}
![{\displaystyle p,q\in A[t]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c1a2e0a3e71f7b320f81e4090bb53df3fd3ce9d)
![{\displaystyle (A[t],\|\!|\cdot |\!\|_{\mathcal {A}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce37a3f21b3061f5ff6c81bd2bf9b4efc318475a)
![{\displaystyle A[t]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13daf73f25e64310e7e5d67c5aa8dd4f97cb884f)
![{\displaystyle p\in A[t]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c749fafdbe6240e74f3f5a2a0b3ad5d59f2ab895)
![{\displaystyle p\not =0_{_{A[t]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec8df52273877cc876d7f084ea5ab48aaa32cbe1)
