In Bruce Blackadder's Book K-Theory for Operator Algebras, The very first definition in the book is as follows:
Definition : A vector bundle over X is a topological space $E$, a continuous map $p: E \rightarrow X $, and a finite dimensional vector space structure on each $E_{x}=p^{-1}(x)$ such that $E$ is locally trivial.
My question is, where is the vector space? I am unsure of how to think about it. On wikipedia, and in the book the first example is the Möbius strip
(see here : https://en.wikipedia.org/wiki/Vector_bundle)
is the vector space the space in which the vector bundle lives or something else?
Thanks in advance.