Intuitive implication of the fact that dual of $L_{p}$-norm space is $L_{q}$-norm space where $\frac{1}{p}+\frac{1}{q}=1$.

Question

While studying the inverse problem theory (I am mainly concerning discrete variables), I learned the theorem that "the dual of $L_{p}$ where $1<p<\infty$ is $L_{q}$ provided that $\frac{1}{p}+\frac{1}{q}=1$". However, I am not acquainted with the concept of duality and I have hardship understanding the intuitive meaning of this theorem. Can somebody explain physical or geometrical meaning of this theorem?

Answer 1

The result is purely analytic, I dont see a way to show it "geometrically" as there is no clear geometry involved in the definition of a $L_p$ space. Think that mathematics is "the science of the un-intuitive", as if mathematics would be intuitive it will be trivial and we will develop it much more faster than we did.

In this context that $L_q(\mu)$ is the dual of $L_p(\mu)$ means that any continuous linear functional in $L_p(\mu)$ is of the form

$$ f\mapsto \int fg \mathop{}\!d \mu $$

for some $g\in L_q(\mu)$.