Duality of Vector Spaces

from Chapter IV, Section 3, Duality of Vector Spaces, Barvinok

Given E, F vector spaces, a non-degenerate bilinear form is called a "duality" of E and F. Examples.

Euclidean space $\langle y,x \rangle = \sum x_i y_i$
Spaces of Symmetric Matrices $\langle A,B \rangle = tr(AB)$
Spaces $L 1$ and $L^\infty$ , $\langle f,g \rangle = \int_0^1 f(s) g(s) ds$ similarly can define dualities between $L p$ and $L q$
Spaces of continuous functions and spaces of signed measures. Let $E = C (0,1)$ be the space of all continuous functions on the interval (0,1) and let $F = V [0,1]$ be the space of signed Borel measures, and let $<f,\mu> = \int_0^1 f d\mu$

Given a duality we can define the notion of duality (or adjoint) linear transformations on the spaces. We say A and $A *$ are dual (or adjoint) linear transformations if

$\langle A(e), f \rangle = \langle e, A^* f \rangle, \quad \text{ for all } e \in E, f \in F$

Duality of Vector Spaces - Adam Oberman Math

Duality of Vector Spaces

Duality of Vector Spaces