Functions: $x\mapsto a\cos x+b\sin x+ c$ where $a,b,c$ vary over all real numbers.
I read that for checking if a set of vectors ($V$) constitute a vector space, for all $u,v \in V$ for every $a,b$, $au+bv\in V$. How can I extend this to real functions?
If you take two functions
$$f_1(x)=a_1\cos x+b_1\sin x+c_1$$
$$f_2(x)=a_2\cos x+b_2\sin x+c_2$$
of your set of real functions $V$, then it is easy to check that for all $\alpha,\beta\in \mathbb R$:
$$\alpha f_1+\beta f_2=(a_1+a_2)\cos x+(b_1+b_2)\sin x+(c_1+c_2)\in V.$$
So it is a vector space.