# pre hilbert space

i have got a question about pre hilbert space. In the lecture, we said that a vector space E with fixed inner product 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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