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?