Consider an arbitrary $f \in L^2 [0,1]^+ $
where $L^2[0,1]^+$ is the function space of square integrable non negative functions.
We say $T$ is an Integral Operator if $T$ is of the form , $$ T(f) = \int_{[0,1]} f(x) K(x,y) dx $$
It's well known that the Fourier Transform constitutes a bijective isometry integral transform between $L^2[\mathbb{R}]$ and $L^2[\mathbb{R}]$
Is there something similiar for compact intervals like $[0,1]$ ? Specifically I am wondering if there is an injective transform with a continuous inverse.
If it makes things easier is there such a transform for $C[0,1]^+$ ?
Note , I use the $+$ primarily since it might be easier to find an operator for non negative functions.