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I have heard the term "space" in context of linear algebra for many years.

In a recent lecture on convex sets and functions, a lecturer was setting foundational knowledge typical for this subject on the defines for linear operations and inner products. He specifically said he was going to define the inner product in $\mathbb{R}^n$ making it an inner product "space". This made me think there was a nuance to the word space that I haven't grasped.

Can someone define what a space is in this context and what we are defining a inner product space rather than treating the inner product as an operation on a set. (Sadly I have not learned real analysis yet, so this question may be trivial) The statement is at the 2:15 mark in this video.

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  • $\begingroup$ In this context, the professor is talking about "topological spaces." Linear algebra is about "vector spaces." Very different concepts. $\endgroup$
    – Thomas Andrews
    Mar 18 at 14:52
  • $\begingroup$ @ThomasAndrews I disagree with the assertion that these are "very different concepts," at least as they pertain to what a "space" is. I cannot think of an example of a mathematician using the word "space" where it doesn't mean something like "a set with some additional structure". A topological space is a set, together with a collection of special sets (the open sets); a vector space is a set (or, really, two sets: vectors and scalars) together with some rules describing how elements of the sets interact. "Space" = "A set with structure". $\endgroup$
    – Xander Henderson ♦
    Mar 18 at 14:59
  • $\begingroup$ But groups and rings are sets with structures, too. Why we call these two structures "spaces" is not just because they are sets with structures, @XanderHenderson $\endgroup$
    – Thomas Andrews
    Mar 18 at 15:03
  • $\begingroup$ @ThomasAndrews I did not say that every set with extra structure is a space. I said that every space (in my experience) is a set with extra structure. $\endgroup$
    – Xander Henderson ♦
    Mar 18 at 15:07
  • $\begingroup$ @XanderHenderson sure, but you said that the fact that they are sets with structures makes them similar. But the world of "sets with structure" is huge. Being both in that world doesn't make them close. $\endgroup$
    – Thomas Andrews
    Mar 18 at 15:27

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