Let, $C[0, 1]$ be the real vector space of all continuous real valued functions on $[0, 1]$, and let $T$ be the linear operator on $C[0, 1]$ given by $$(Tf)(x) =\int_{0}^{1}\sin(x + y)f(y) dy,\quad x\in[0, 1].$$ Then what is the dimension of the range space of $T\;?$
My attempt:
I know that $C[0, 1]$ is a vector space of infinite dimension. But I don't know what is basis of $C[0, 1]$, that's why could not find the matrix representation. I've read this problem in MSE. But I'm unable to solve the problem. Any hints will be apprciated.
Note that $$\sin(x+y) = \sin(x) \cos(y) + \sin(y) \cos(x).$$ Using this, we see that for any $f \in C[0,1]$, $$(Tf)(x) = \left(\int^1_0 \cos(y) f(y) dy \right) \sin(x) + \left(\int^1_0 \sin(y) f(y) dy \right) \cos(x).$$ Thus the image of any continuous function under $T$ will be a linear combination of two functions: $\sin(x)$ and $\cos(x)$. This shows that the range of $T$ has dimension $2$.