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?