I am a beginner in functional analysis. Recently, I came across the term "subdifferential" of a convex function. I would like to know the notion of subdifferential of a convex function in detail (or some references). In particular, if a real valued convex function has minimum value $0$ at $0$, then how to explain it using the subdifferential of that function?
Thanks in advance.
By defintion, subdifferential $\partial f(x_0)$ of $f(x)$ at point $x_0$ consists of vectors $v$ such that $f(x) - f(x_0) \geq (v, x-x_0)$ for all $x \in X$, where $(\cdot, \cdot)$ denotes dot product. If $x^*$ is argmin of $f(x)$, then $\forall x \in X :f(x) \geq f(x^*) \Leftrightarrow f(x) - f(x^*) \geq 0 = (0, x - x^*)$, so $0 \in \partial f(x^*)$.
Moreover, the opposite statement is also true: if $0 \in \partial f(x^*)$, then $f(x) \geq f(x^*)$ for all $x \in X$, so $f$ has minimum at $x^*$.