Differences
This shows you the differences between two versions of the page.
hilbert_space [2013/09/13 19:32] nikolaj |
hilbert_space [2014/03/21 11:11] |
||
---|---|---|---|
Line 1: | Line 1: | ||
- | ===== Hilbert space ===== | ||
- | ==== Definition ==== | ||
- | | @#88DDEE: $V$ | | ||
- | | @#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 ==== | ||
- | === Reference === | ||
- | Wikipedia: [[http://en.wikipedia.org/wiki/Hilbert_space|Hilbert space]] | ||
- | ==== Parents ==== | ||
- | === Subset of === | ||
- | [[Pre-Hilbert space]] | ||
- | === Requirements === | ||
- | [[Cauchy sequence]] |