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norm [2013/09/06 23:28] nikolaj |
norm [2016/05/01 16:01] nikolaj |
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| ===== Norm ===== | ===== Norm ===== | ||
| - | ==== Definition ==== | + | ==== Set ==== |
| - | | @#88DDEE: $F$ ... subfield of $\mathbb{C}$ | | + | | @#55CCEE: context | @#55CCEE: $F$ ... subfield of $\mathbb{C}$ | |
| - | | @#88DDEE: $V$ ... $F$-vector space | | + | | @#55CCEE: context | @#55CCEE: $V$ ... $F$-vector space | |
| + | | @#FFBB00: definiendum | @#FFBB00: $p\in \mathrm{Norm}(V)$ | | ||
| + | | @#55EE55: postulate | @#55EE55: $p:V\to \mathbb R $ | | ||
| + | | $v,w\in V$ | | | ||
| + | | @#55EE55: postulate | @#55EE55: $p(v+w) \le p(v)+p(w)$ | | ||
| + | | @#55EE55: postulate | @#55EE55: $p(v)=0 \implies v=0$ | | ||
| + | | $\lambda\in F$ | | | ||
| + | | @#55EE55: postulate | @#55EE55: $p(\lambda\cdot v) = |\lambda|\cdot p(v)$ | | ||
| - | | @#FFBB00: $p\in \mathrm{Norm}(V)$ | | + | ----- |
| - | + | === Discussion === | |
| - | | @#55EE55: $p:V\to \mathbb R $ | | + | |
| - | + | ||
| - | | $v,w\in V$ | | + | |
| - | + | ||
| - | | @#55EE55: $p(v+w) \le p(v)+p(w)$ | | + | |
| - | | @#55EE55: $p(v)=0 \implies v=0$ | | + | |
| - | + | ||
| - | | $\lambda\in F$ | | + | |
| - | + | ||
| - | | @#55EE55: $p(\lambda\cdot v) = |\lambda|\cdot p(v)$ | | + | |
| - | + | ||
| - | ==== Discussion ==== | + | |
| ^ $ p(v)\ge 0 $ ^ | ^ $ p(v)\ge 0 $ ^ | ||
| The last axiom $\ p(v)=0 \implies v=0\ $ isn't part of [[seminorm]]. | The last axiom $\ p(v)=0 \implies v=0\ $ isn't part of [[seminorm]]. | ||
| + | |||
| === Reference === | === Reference === | ||
| Wikipedia: [[http://en.wikipedia.org/wiki/Norm_%28mathematics%29|Norm]] | Wikipedia: [[http://en.wikipedia.org/wiki/Norm_%28mathematics%29|Norm]] | ||
