I have a basic clarificatory question about the definition of a subgradient from the book Infinite Dimensional Analysis by Aliprantis and Border. Here is the definition.
Given a dual pair $\langle X,X' \rangle$ and a real-valued convex function $f$ on $X$, we say that $x' \in X'$ is a subgradient of $f$ at $x$ if $f(y) \geq f(x) + x'(y-x)$ for all $y \in X$.
And here is the definition of dual pair.
A dual pair is a pair $\langle X,X' \rangle$ of vector spaces with a bilinear functional $(x,x') \mapsto \langle x,x'\rangle$ from $X \times X'$ to $\mathbb R$ that separates the points of $X$ and $X'$. That is, $(x,x') \mapsto \langle x,x'\rangle$ is linear in both of its arguments and satisfies (1) if $\langle x,x' \rangle = 0$ for each $x' \in X'$, then $x = 0$, and (2) if $\langle x,x' \rangle = 0$ for each $x \in X$, then $x'=0$.
My question (again, very basic) is how to understand the expression $x'(y-x)$ in the definition of subgradient. Is this just an alternative notation for $\langle y-x, x' \rangle$? If not, then I am unsure how to connect the definition of the subgradient to the definition of dual pairs.
Yes, this is just an alternative notation. Since the dual space is frequently viewed as all bounded linear functionals from $X$ to $\mathbb{R}$, sometimes the pairing $\langle x, x'\rangle$ is defined by the "evaluation" of $x'$ at $x$, written $\langle x, x'\rangle = x'(x)$.
The definition from the book uses an alternative (albeit equivalent) definition, where $x'\in X'$ is a vector. In this case, the bounded linear functional that $x'$ "represents" is $x\mapsto\langle x, x'\rangle$. However, folks frequently abuse notation and do not distinguish between the vector $x'\in X'$ and its functional form, which can lead to the (somewhat confusing) convention that $x'(x)=\langle x,x'\rangle$.