infinite vector spaces,

In general, how does one prove that a vector space is infinite dimensional?
My first attempt would be to suppose it were finite, then there exists a
finite basis, and then show that there is some vector in the vector space
that cannot be written as a linear combination of those finite basis
vectors. But how would you do this for say the set of all real-valued
functions, for example?I've only learned the trigonometric bases for
functions from Fourier series, so it is difficult for me to make progress
with my idea.


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