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I have always thought that linear and vector space is the same thing (for example on Wikipedia, it refers to the same).

However, in Van Mill´s book "The Infinite-Dimensional Topology of Function Spaces", the author claims that "vector space is an algebraic structure which may, or may not carry a topology while linear space is automatically a topological space".

What do you think about this?

What is the difference between vector and linear space then? (Apart from the topology, obviously.)

Does it mean that vector space is generalization of linear space?

Could you give any example of a vector space that is not linear?

Thank you.

My idea:

The topology on linear space is given by continuity of the operations of addition of vectors and multiplication of vector and scalar. So if the space is not linear, but just vector one, these operations are still there, but may not be continuous.

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    $\begingroup$ Did you find the definition of linear space in that book? $\endgroup$ Commented Aug 27, 2021 at 11:18
  • $\begingroup$ @ArcticChar Yes, it is a real vector space L, carrying a (separable metrizable) topologx with the property that the algebraic operations of addition and scalar multiplication are continuous. $\endgroup$ Commented Aug 27, 2021 at 11:20
  • $\begingroup$ I've always understood them to be the same, and have been told by a professor (one who should know) that they are the same. While the term "normed linear space" is more common than "normed vector space", I have seen both, and I've never seen anyone refer to a "topological linear space", only a "topological vector space". $\endgroup$ Commented Aug 27, 2021 at 11:20
  • $\begingroup$ @TheoBendit Your last sentence makes sense tho (if they are really different), because if the linear space is already topological, it is useless to call it "topological linear space". While with vector space, it is not clear that it is topological, thats why "topological vector space" is used. $\endgroup$ Commented Aug 27, 2021 at 11:21
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    $\begingroup$ When you write a book, you can make any definitions you want, as long as you make them clear. I think Van Mill has done that. I suspect the reason is that the book is mostly going to deal with vector spaces with a separable metrizable topology, and the author doesn't want to write "vector space with a separable metrizable topology" several hundred times, so the author just uses the term "linear space" as a convenient abbreviation for that. No harm, no foul. $\endgroup$ Commented Aug 27, 2021 at 12:03

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I honestly think trying to encode the topology into the terminology "linear space" is weird. I've always seen "vector space" and "linear space" used synonymously, and I think these are appropriately named sense since "vector space" suggests we're talking about a set of vectors (and vectors are to be added and scalar-multiplied), and "linear space" suggests a set with some structure of "linearity". So, in every situation I've seen, they just refer to the purely algebraic structure $(V,+,\cdot, \Bbb{F})$.

If we want to talk about a structure on top of the algebraic structure alone, then our terminology should reflect that explicitly. For example, if we wish to talk about a topology on top of the algebraic structure, we should speak of "topological vector space" or "topological linear space" (I've actually never seen this second term before, but I am sure people will know exactly what you mean). Furthermore we may then wish to talk about further structure, such as having a norm, in which case we'd speak of "normed vector space" or "normed linear space". If we wish to specify further properties, such as completeness, then we'd speak of "complete topological vector space" or "complete topological linear space" also one talks about "complete normed vector space" or "complete normed linear space" (i.e Banach spaces). We can also add about other adjectives, such as "complete, locally-convex, metrizable, topological vector/linear space" (i.e Frechet space).

Trying to unnecessarily suppress adjectives, and subsume them into a definition, is just weird to me (especially when it is already extremely common to use the terms "vector space" and "linear space" synonymously). Anyway, if your book defines things in one way, fine, but just know that it doesn't hurt to be more explicit in the terminology especially when you're communicating with others (after all, a large part of math is about communicating ideas).

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  • $\begingroup$ In Hatcher's Algebraic topology, every map is continuous. $\endgroup$ Commented Aug 27, 2021 at 16:00
  • $\begingroup$ @Arctic Char hence my last sentence in the last paragraph $\endgroup$
    – peek-a-boo
    Commented Aug 27, 2021 at 19:35

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