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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... – GEdgar Nov 23 '12 at 19:31

There are many possible inner products you can put on a vector space $V$. If you choose one $\langle \cdot \rangle$, then $(V, \langle \cdot \rangle)$ is a pre-Hilbert space (or an inner-product space).

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It means that in principle there are many possible inner products that you can define on a vector space. So one has to be chosen when you want to consider the space together with the inner product structure.

To make a simple example, consider $E=\mathbb R^2$. Then you have the canonical inner product $$ (x_1,y_1)\cdot(x_2,y_2)=x_1x_2+y_1y_2. $$ But you can also define $$ (x_1,y_1)\odot(x_2,y_2)=5x_1x_2+93y_1y_2. $$ Both are inner products, and you can easily tell from the second example that there are infinitely many others.

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