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seminorm [2014/03/21 11:11] 127.0.0.1 external edit |
seminorm [2016/05/01 15:57] nikolaj |
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| | @#55CCEE: context | @#55CCEE: $F$ ... subfield of $\mathbb{C}$ | | | @#55CCEE: context | @#55CCEE: $F$ ... subfield of $\mathbb{C}$ | | ||
| | @#55CCEE: context | @#55CCEE: $V$ ... $F$-vector space | | | @#55CCEE: context | @#55CCEE: $V$ ... $F$-vector space | | ||
| - | |||
| | @#FFBB00: definiendum | @#FFBB00: $p\in \mathrm{SemiNorm}(V)$ | | | @#FFBB00: definiendum | @#FFBB00: $p\in \mathrm{SemiNorm}(V)$ | | ||
| - | |||
| | @#55EE55: postulate | @#55EE55: $p:V\to \mathbb R $ | | | @#55EE55: postulate | @#55EE55: $p:V\to \mathbb R $ | | ||
| - | + | | $v,w\in V$ | | | |
| - | | $v,w\in V$ | | + | |
| | @#55EE55: postulate | @#55EE55: $p(v+w) \le p(v)+p(w)$ | | | @#55EE55: postulate | @#55EE55: $p(v+w) \le p(v)+p(w)$ | | ||
| - | + | | $\lambda\in F$ | | | |
| - | | $\lambda\in F$ | | + | |
| | @#55EE55: postulate | @#55EE55: $p(\lambda\cdot v) = |\lambda|\cdot p(v)$ | | | @#55EE55: postulate | @#55EE55: $p(\lambda\cdot v) = |\lambda|\cdot p(v)$ | | ||
| ==== Discussion ==== | ==== Discussion ==== | ||
| - | A [[Norm]] is a seminorm with the adition axiom | + | A [[Norm]] is a seminorm with the adition axiom |
| + | |||
| + | $p(v)=0 \implies v=0$ | ||
| - | $p(v)=0 \implies v=0$ | + | (which I also write as $p(!0)=0$.) |
| + | |||
| === Reference === | === Reference === | ||
| Wikipedia: [[http://en.wikipedia.org/wiki/Norm_%28mathematics%29|Norm]] | Wikipedia: [[http://en.wikipedia.org/wiki/Norm_%28mathematics%29|Norm]] | ||
