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I have got a question about pre-Hilbert space. In the lecture, we said that a vector space E with a fixed inner product $\langle \cdot,\cdot\rangle$ is called pre-Hilbert space.

My question is, what does "fixed" mean here? Does this mean that all vectors in this space have the same inner product? I appreciate if someone can help. Thanks!

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    One wonders why they didn't also say it is a *fixed* vector space...2012-11-23

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