Extension of $L^p(\mathbb{R}^n)$ to a metric space

Question

I know the following definition of $L^p(\mathbb{R}^n)$ with $1\leq p < \infty$: $$L^p(\mathbb{R}^n) := \left\{ f:\mathbb{R}^n \to \mathbb{C} : \left( \int_{\mathbb{R}^n}|f|^pd\mu \right)^{1/p} < \infty \right\}.$$ If we consider the norm $$\|f\|_p = \left( \int_{\mathbb{R}^n}|f|^pd\mu \right)^{1/p}$$ for all $f\in L^p(\mathbb{R}^n)$, we can see that this norm does not induce a metric. Therefore, we have to build an equivalence relation to get a metric space.

Consider the following equivalence relation on $L^p(\mathbb{R}^n)$: $$f\sim g \quad \text{if and only if}\quad\|f-g\|_p=0.$$ Consider the following function from $(L^p(\mathbb{R}^n)/\sim)^2$ to $\mathbb{R}$: $$d([f]_{\sim},[g]_\sim) = \|f-g\|_p.$$ How can I prove that this function is well-defined? In other words, how can I verify that if $f_1\sim f_2$ and $g_1\sim g_2$ then $\|f_1-g_1\|_p = \|f_2-g_2\|_p$?

Answer 1

Use Minkowski's inequality: $$|\|f_1-g_1\|_p-\|f_2-g_2\|_p|\leq \|(f_1-g_1)+(g_2-f_2)\|_p\leq \|f_1-f_2\|_p+\|g_1-g_2\|_p=0$$