Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision Next revision Both sides next revision | ||
|
hilbert_space [2013/09/06 22:04] 127.0.0.1 external edit |
hilbert_space [2013/09/13 19:37] nikolaj |
||
|---|---|---|---|
| Line 1: | Line 1: | ||
| ===== Hilbert space ===== | ===== Hilbert space ===== | ||
| ==== Definition ==== | ==== Definition ==== | ||
| - | | @#88DDEE: $V$ | | + | | @#FFBB00: $\mathrm{Hilbert}(V)\equiv \mathrm{PreHilbert}(V)\cap \mathrm{Banachspace}(V)$ | |
| - | + | ||
| - | | @#FFBB00: $\mathrm{Hilbert}(V)$ | | + | |
| - | + | ||
| - | | @#88DDEE: $\mathrm{Hilbert}(V)\subseteq \mathrm{PreHilbert}(V)$ | | + | |
| - | + | ||
| - | | $\mathcal V\in \mathrm{Hilbert}(V)$ | | + | |
| - | | $v_\infty \in \mathcal V $ | | + | |
| - | | $v\in \mathrm{CauchySeq}(V)$ | | + | |
| - | + | ||
| - | 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 === | ||
| Line 20: | Line 7: | ||
| ==== Parents ==== | ==== Parents ==== | ||
| === Subset of === | === Subset of === | ||
| - | [[Pre-Hilbert space]] | + | [[Pre-Hilbert space]], [[Banach space]] |
| - | === Requirements === | + | |
| - | [[Cauchy sequence]] | + | |
