General form of bounded linear functionals on Banach spaces

Question

It is a famous result due to Riesz that every bounded linear functional $f$ on a Hilbert space $\mathcal{H}$ is of the form $f(x)=\langle x,z \rangle$ for a unique $z\in H$.

On p.188 of Introductory Functional Analysis and Applications, Erwin Kreyszig mentions that:

It is of practical importance to know the general form of bounded linear functionals on various spaces. ... For general Banach spaces such formulas and their derivation can sometimes be complicated.

Could someone give a brief statement of some results concerning the form of bounded linear functionals on Banach spaces? What might be some good resources to get an overview of results addressing this issue? Thanks!

Answer 1

For example: For the real Banach space $L^p(\mathbb R)$, with $1 < p < \infty$, the "conjugate space" is $L^q(\mathbb R)$ where $\frac{1}{p}+\frac{1}{q}=1$. For general linear functional $T$ on $L^p$ there exists $g \in L^q$ so that
$$ T(f) = \int_{\mathbb R} f(x) g(x)\;dx\qquad\text{for all }f \in L^p . $$ Of course, in case $p=2$, then also $q=2$ and $\langle f, g \rangle :=\int fg$ is the inner product.

A reference for many Banach spaces and their conjugate spaces:

Dunford, Nelson; Schwartz, Jacob T., Linear operators. I. General theory. (With the assistence of William G. Bade and Robert G. Bartle), Pure and Applied Mathematics. Vol. 7. New York and London: Interscience Publishers. xiv, 858 p. (1958). ZBL0084.10402.