For all $f \in E$, we set, $$\forall s \in [0,1], \quad T(f)(s) = \int_0^1 k_s(t) f(t)\,\mathrm{d}t$$ Determine the eigenvalues of $T$ and show that the associated eigenspaces are one-dimensional.
For all $f \in E$, we set,
$$\forall s \in [0,1], \quad T(f)(s) = \int_0^1 k_s(t) f(t)\,\mathrm{d}t$$
Determine the eigenvalues of $T$ and show that the associated eigenspaces are one-dimensional.