a set whose elements made of a field equipped with additive and scalar multiplicative operations satisfying some axioms related to linearity (or vector space axioms)
intuition
What commonly pops out of our heads when we hear vectors are arrays of numbers.
In fact, it is an example of vector space, known as finite dimensional vector space.
Sequences also form a countably infinite dimensional vector space.
And functions from complete domain forms an uncountably infinite dimensional vector space.