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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?

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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.

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