Prove an identity without using Hölder's inequality

Question

How to prove the following without using Hölder's inequality :

$$ \|f\|_{p} = \sup_{\|g\|_q =1} \int |fg| d\mu ; \frac{1}{p} + \frac{1}{q} =1$$

Answer 1

I'll try to clarify what Wikipedia says:

The Minkowski inequality is the triangle inequality in $L^p(S)$. In fact, it is a special case of the more general fact $$\|f\|_p = \sup_{\|g\|_q = 1} \int |fg| d\mu, \qquad 1/p + 1/q = 1 \tag{*}$$ where it is easy to see that the right-hand side satisfies the triangular inequality.

Indeed, once you have (*), Minkowski's inequality easily follows: $$\|f+h\|_p = \sup_{\|g\|_q = 1} \int |(f+h)g| d\mu \le \sup_{\|g\|_q = 1} \int |fg| d\mu+\sup_{\|g\|_q = 1} \int |hg| d\mu =\|f\|_p+\|h\|_p $$

As Daniel Fischer said, (*) is a statement that contains Hölder's inequality. If you are proving (*), you are proving Hölder's inequality along the way.

If you want a proof of Minkowski's inequality without Hölder's inequality, see here.