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In this question, the word "generate" is used when the author of the book mentioned a vector space. And also, in a Gowers's article, it is used in "vector space generated by the functions $[v,w]$". There is another similar concept called "linear span" in linear algebra. In Gowers's article, he also uses the word "span" as "the space spanned by the functions [v,w]".

Here is my question:

Are these two concepts the same? Or what's the difference?

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    "Generate" is a verb; we talk about vectors "generating" something or other (here, a subspace). "Linear span" is a (compound) noun. The linear span of a set of vectors is precisely the subspace that set of vectors generate or that they "span" ('to span' is a verb, 'span' is a noun, so "span" can be used in both senses). – Arturo Magidin Jul 12 '11 at 19:16

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"Span" is less ambiguous and "generate" is more general.

In the contexts of vectors in a vector space, "generated by" and "spanned by" mean the same thing. However, "generate" can mean various other things (for example generation as a module, as an algebra, as a field, etc.) if there is extra structure on the vector space.

A general definition of generation is as follows: let $e_i$ be elements of some structured set $A$. A subset $B \subset A$ is said to be the structure generated by the $e_i$ if it is the intersection of all substructures of $A$ containing the $e_i$. (The key word here is "substructure": the notion of generation changes depending on what kind of structure one is considering on $A$.) Of course this definition only makes sense if the intersection of substructures of $A$ is another substructure, but this is in my experience true of structures to which the word "generate" are applied.

Qiaochu Yuan
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    To bypass your last caution, can't we just say that $B$ is generated by the $e_i$ if $B$ is the smallest substructure containing them? (assuming there is a notion of 'smallest', probably just set inclusion?) – wildildildlife Jul 12 '11 at 19:45
  • @wildildildlife: there is no guarantee that there is a notion of "smallest" in general (the poset of substructures containing the $e_i$ may have more than one minimal element, or perhaps none!) One sufficient condition that guarantees this is that the intersection of substructures is a substructure. Fortunately, as I said, in my experience this assumption is satisfied in every situation in which people use the word "generate." – Qiaochu Yuan Jul 12 '11 at 19:48
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    Mac Lane gives the following definition for ‘span’: let $\mathscr{U} : \mathbf{C} \to \mathbf{D}$ be a functor, and $c$ an object of $\mathbf{C}$; then an arrow $f : d \to \mathscr{U} c$ spans if there are no proper monomorphism $i : c' \rightarrowtail c$ such that $f$ factors through $\mathscr{U} i$. It's essentially the same thing as the intersection of substructures, except phrased in a way which doesn't require the (set-theoretic) intersection of substructures to be another substructure (which will happen anyway if $\mathbf{D} = \textbf{Set}$ and $\mathscr{U}$ creates all pullbacks). – Zhen Lin Jul 13 '11 at 01:11
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Actually, the intersections of substructures of the same type should be a substructure of that same type. If i've not misunderstood the content, which is not unlikely, this should be the theorem which proves it (literal quote from "Algebra, Vol 1 - L. Redei - Pergamon Press, 1967 pag 74"):

"Theorem 46. Let S denote an A-structure and Sl, S2,... denote B-sub- structures of S where A, B is a suitable pair of systems of structure axioms for semigroups, groups, modules, rings or skew fields. If the intersection D = S1 inters. S2 inters. ... is not empty, this is also a B-substructure of S, except in the case when A denotes the ring axioms, B the skew field axioms and D is the zero element of S."

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