can you help me to solve this exercice? Show that the kernel of the operator $M:L^2((-1,1))\to L^2((-1,1))$ has infinite dimension. The operator is: $$ M[f](x):= \int_{(-1,1)} \sin(xy)f(y)\,\,\,dy. $$
Hint. Denote with P the operator that $f(x)\to f(-x)$, we have $MP[f]=PM[f]$. Observe how M operates with even functions.
If $f$ is an even function, then $M(f) = 0.$ QED