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hilbert_space [2013/09/06 22:04] 127.0.0.1 external edit |
hilbert_space [2013/09/13 19:32] nikolaj |
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| | @#88DDEE: $V$ | | | @#88DDEE: $V$ | | ||
| - | | @#FFBB00: $\mathrm{Hilbert}(V)$ | | + | | @#FFBB00: $\mathcal V \in \mathrm{Hilbert}(V)$ | |
| - | | @#88DDEE: $\mathrm{Hilbert}(V)\subseteq \mathrm{PreHilbert}(V)$ | | + | | @#88DDEE: $\mathcal V \in \mathrm{PreHilbert}(V)$ | |
| - | | $\mathcal V\in \mathrm{Hilbert}(V)$ | | ||
| - | | $v_\infty \in \mathcal V $ | | ||
| | $v\in \mathrm{CauchySeq}(V)$ | | | $v\in \mathrm{CauchySeq}(V)$ | | ||
| + | | @#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: | 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: | ||
