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hilbert_space [2013/09/13 19:32] nikolaj |
hilbert_space [2013/09/13 19:37] nikolaj |
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| ===== Hilbert space ===== | ===== Hilbert space ===== | ||
| ==== Definition ==== | ==== Definition ==== | ||
| - | | @#88DDEE: $V$ | | + | | @#FFBB00: $\mathrm{Hilbert}(V)\equiv \mathrm{PreHilbert}(V)\cap \mathrm{BanachSpace}(V)$ | |
| - | + | ||
| - | | @#FFBB00: $\mathcal V \in \mathrm{Hilbert}(V)$ | | + | |
| - | + | ||
| - | | @#88DDEE: $\mathcal V \in \mathrm{PreHilbert}(V)$ | | + | |
| - | + | ||
| - | | $v\in \mathrm{CauchySeq}(V)$ | | + | |
| - | | @#DDDDDD: $v_\infty \in \mathcal V $ | | + | |
| - | + | ||
| - | The space $\mathcal V$ is complete: For each Cauchy sequence $(v)_{i\in\mathbb N}$, there is a limit $v_\infty\in\mathcal V$ w.r.t. the natural norm: | + | |
| - | + | ||
| - | | @#55EE55: $\exists v_\infty.\ \mathrm{lim}_{n\to\infty}\Vert v_n-v_\infty \Vert = 0$ | | + | |
| ==== Discussion ==== | ==== Discussion ==== | ||
| === Reference === | === Reference === | ||
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| ==== Parents ==== | ==== Parents ==== | ||
| === Subset of === | === Subset of === | ||
| - | [[Pre-Hilbert space]] | + | [[Pre-Hilbert space]], [[Banach space]] |
| - | === Requirements === | + | |
| - | [[Cauchy sequence]] | + | |
