Let $K: [0,1] \times \mathbb{R}^n \to \mathbb{C}$ have the properties:
- $K(x,\cdot) \in L^2(\mathbb{R}^n)$ for all $x\in[0,1]$
- For every $f\in L^2(\mathbb{R}^n)$ the function $ x\mapsto \int_{\mathbb{R}^n} K(x,y)f(y)\,dy$ is continuous on $[0,1].$ Prove that the intergral operator ${\bf K}$ defined by $ {\bf K}f(x) = \int_{\mathbb{R}^n} K(x,y)f(y)\,dy$ is bounded from $L^2(\mathbb{R}^n)$ to $C([0,1])$.
I know that $L^2(\mathbb{R}^n)$ and $C([0,1])$ are Banach spaces. So maybe we could apply the closed graph theorem? Then closed $\implies$ continuous $\implies$ bounded. How could I prove it? Or should I do it some other way?