Linear map from $L^1 \rightarrow L^{\infty}.$

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I was wondering how I can show that any linear map $T: L^1(\Omega) \rightarrow L^{\infty}(\Omega)$ can be represented as an integral operator

$$T(f)(x):=\int_{\Omega} K(x,y)f(y) dy.$$

Does anybody know how to show this or where this follows from?

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Here you will find your theorem, a sketch of its proof, some historical remarks and a generalisation.