Prove that $L^1$ is a Banach algebra with multiplication defined by convolution

Question

To be more specific, prove that $L^1(\mathbb{R}^n)$ with multiplication defined by convolution: $$ (f\cdot g)(x)=\int_\mathbb{R^n}f(x-y)g(y)dy $$ is a Banach algebra. All the properties of Banach algebra are easily proved except the last one: $$ \|f\cdot g\|\le \|f\|\|g\|. $$ Could anyone give me a hint?

Actually I found a proof on the web, but I don't see why the last equality holds. $$\|f* g\|_1 = \int |f*g(x)|dx \le \iint |f(x-y) g(y)|dydx = \|f\|_1 \|g\|_1$$

Answer 1

\begin{align} \iint |f(x-y)g(y)|\;dx\;dy &= \int\left[\int|f(x-y)|\;dx\right]|g(y)|\;dy \\ &= \int\left[\int|f(x)|\;dx\right]|g(y)|\;dy \\ &= \left[\int|f(x)|\;dx\right]\;\left[\int |g(y)| \;dy\right]\\ & = \|f\|_1\;\|g\|_1 \end{align}