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.
Best Answer
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).