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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.

Dominic Petti
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    When I informally think of a vector bundle I usually imagine a base manifold which is some sort of curved sheet and a cylinder above it, which is meant to be a local trivialization. Now imagine to divide this cylinder up into disjoint fibers, straight lines that do not intersect each other and project onto different points on the base manifold. These fibers are the vector spaces $E_x$. Of course this is not a faithful representation of the geometry, because each fiber is one-dimensional, but it usually works for my intuition. – Gibbs Mar 30 '20 at 20:51
  • @Gibbs Thanks! I appreciate it – Dominic Petti Mar 30 '20 at 20:52
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    You should see some examples. The best one is the tangent bundle of a manifold. – Thomas Rot Mar 31 '20 at 09:05

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