Let's see: Let A be a non-empty set, N the set of positive whole numbers, and X the set of all functions f
f: A --> N
with usual notation.
Assume as is common, the scalars are the set of real numbers, but the set of complex numbers will also do.
So, is X a vector space and, thus, each f in X a vector?
No, since -f is not in X. Neither is (1/2)f.
Some references (with TeX markup):
Paul R.\ Halmos, {\it Finite-Dimensional Vector Spaces, Second Edition\/}
linear algebra treated as functional analysis.
Walter Rudin, {\it Real and Complex Analysis\/}
with Lebesgue integration and, then, Banach and Hilbert vector spaces.
Walter Rudin, {\it Functional Analysis\/}
with Fourier theory.
Jacques Neveu, {\it Mathematical Foundations of the Calculus of Probability\/}
with random variables, that is, functions from a probability space to, usually, the set of real numbers with convergence results, building on the work A. Kolmogorov building on the work of H. Lebesgue.