Theorem: Let E be a measurable set, \(1 \le p\le \infty\). Then \((X,\|\cdot\|_p)\) is a normed linear space. In particular, for \(f,g\in L^p(E)\)

$$ \|f+g\|_p\le \|f\|_p+\|g\|_p $$

Proof: I assume all the other properties, like being a linear space, and properties of \(\|\cdot\|_p\) other than the triangle inequality, have been proved earlier. That is to say, the only property we prove is the triangle inequality.

We will apply Holder's inequality (sorry, not going to google the umlaut symbol) using the "conjugate function". The conjugate of a general function \(h\in L^p(E)\) is defined as

$$ h^* = \|h\|_p^{1-p} \text{sgn}(h)\cdot |h|^{p-1} $$

This function is built so as to be \(h\in L^{p^*}(E)\) and satisfy the equations

$$ \int_E hh^* = \|h\|_p $$

and

$$\|h^*\|_{p^{*}} =1$$

In the following chain of inequalities, they are justified by (1) the above, (2) linearity, (3) Holder’s, (4) the above.

$$ \begin{aligned}\int_E (f+g) (f+g)^* &= \|f+g\|_p \\&= \int_E f(f+g)^* + \int_E g(f+g)^* \\&\le \|f\|_p\|(f+g)^*\|_{p^*} + \|g\|_p\|(f+g)^*\|_{p^*} \\&= \|f\|_p+\|g\|_{p^*}\end{aligned} $$