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A vector space $R$ , with fixed inner product $\langle x,y \rangle$ is called pre-Hilbert Space

A vector space $E$ with fixed norm $||.||$ is called Normed Space.

A set $X$ with fixed metrics $d(x,y)$ is called Metric Space.

But I am wondering:

What is the meaning of "fixed" in these three definitions? Why do we need that word in the definitions?

According to Wikipedia, Euclidean Space is a pre-Hilbert Space. But I used to think that in the Euclidean Space, distance between two points $d(x,y)$ is defined also. Is not it? If yes, what is the difference between Metric Space and pre-Hilbert Space?

Can we say that Metric Spaces cover Normed and pre-Hilbert Spaces and Normed Spaces cover pre-Hilbert Spaces?

Also, I would like to ask what is the use of having pre-Hilbert Spaces? I am too ignorant about the subject but now when I think, I feel like if we have a space with more properties like Metric Spaces, then why would we use pre-Hilbert Spaces? Are there situations in applied math or physics that we cannot use Metric Spaces to find the solution and have to use pre-Hilbert Spaces? Could you please give me some information about these three spaces' historical development and their differences?

Regards,

Amadeus

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Every inner product induces a norm $\lVert x\rVert := \sqrt{\langle x,x \rangle}$ and every norm induces a metric $d(x,y):=\lVert x - y\rVert$. So every pre Hilbert Space is a Normed Space is a Metric Space (with the induced norm/metric) –  Stefan Dec 4 '12 at 17:48
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1 Answer

up vote 2 down vote accepted

A space can have more than one metric, norm or inner product, so fixed just means looking at a specific one at a time, nothing more.

An inner product induces a norm and a norm induces a metric in the sense that an inner product once defined generates a norm and a norm once defined generates a metric in a natural manner.

A set can be an inner product space, normed space and metric space all at once, for example $\mathbb R^n$.

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