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Pre-Hilbert space
Definition
| $\langle\mathcal V,\langle\cdot|\cdot\rangle\rangle \in \mathrm{Pre-Hilbert}(V)$ |
| $\mathcal V \in \mathrm{VectorSpace}(V,\mathbb C)$ |
| $\langle\cdot|\cdot\rangle:V\times V\to \mathbb C$ |
| $u,v,w\in V$ |
| $a,b\in \mathbb C$ |
| $\overline{\langle v|w \rangle}=\langle w|v \rangle$ |
| $v \ne 0 \Rightarrow \langle v|v \rangle > 0 $ |
| $v = 0 \Rightarrow \langle v|v \rangle = 0 $ |
| $\langle u|a\cdot v+b\cdot w \rangle = a\cdot \langle u|v \rangle+b\cdot \langle u|w \rangle $ |
| $\langle a\cdot v+b\cdot w | u \rangle = \overline a\cdot \langle v|u \rangle+\overline b \cdot \langle w|u \rangle $ |
Ramifications
Discussion
A vector space is a $F$-module over $V$, where $F$ is a field, not just a ring.
One speaks of an $F$-vector space over $V$. Here $F$ and $V$ are just sets.
Reference
Context
Subset of
Requirements
