Proof $L^{p}(X,\mu)$ is a linear space normed with respect to the norm

Question

I am studying Lp-Spaces and Banach-Spaces and the norm for $ p \in [1, \infty) $ is given by $$||f||_{L_{p}}=\left ( \int _{X} ||f(x))||^{p} \right )^{1/p}$$ but i didn't find how to test it. Do you know any book or pdf where I can find the demo? Thank you so much