I was confused by the Lipschitz-continuity of value function when reading $\textit{Convex and Stochastic Optimization}$ p51 (1.245) by J. Frédéric Bonnans. And this is what (1.245) says:
If $v(y)$ is finite at some $y\in Y$, then$$|v(y^{\prime})-v(y)|\leq \sup_{y^{*}\in Y^{*}_{0}}|\langle y^{*},y^{\prime}-y\rangle|\leq(\sup_{y^{*}\in Y^{*}_{0}}\|y^{*}\|)\|y^{\prime}-y\|.\quad\quad(1.245)$$
More precisely, I was wondering how to get the first inequality in (1.245). And here are some precondition of what (1.245) says.
Let $X$, $Y$ be two Banach space. $X_{0}$, $Y_{0}^{*}$ are nonempty subsets of $X$ and $Y^{*}$ respectively.($Y^{*}$ denotes the dual space of $Y$) $X_{0}$ is closed and $Y_{0}^{*}$ is bounded. $L:X_{0}\times Y_{0}^{*}\rightarrow \mathbb{R}$ satisfies that $L(\cdot,y^{*})$ is l.s.c and convex for each fixed $y^{*} \in Y^{*}_{0}$. $L(x,y^{*})$ is an infimum of $*$affine functions for each fixed $x\in X_{0}$. $v(y):=\inf_{x\in X_{0}}\sup_{y^{*}\in Y_{0}^{*}}\langle y^{*},y\rangle +L(x,y^{*})$.
Hence $$ |v(y^{\prime})-v(y)|\leq \sup_{y^{*}\in Y^{*}_{0}}|\langle y^{*},y^{\prime}-y\rangle|\Leftrightarrow |\inf_{x\in X_{0}}\sup_{y^{*}\in Y_{0}^{*}}\langle y^{*},y^{\prime}\rangle +L(x,y^{*})-\inf_{x\in X_{0}}\sup_{y^{*}\in Y_{0}^{*}}\langle y^{*},y\rangle +L(x,y^{*})|\leq \sup_{y^{*}\in Y^{*}_{0}}|\langle y^{*},y^{\prime}-y\rangle|$$ It can be shown that $v(y)$ is convex but I don't know if it is useful for (1.245).
Can anyone give me a hint or explanation of the first inequality in (1.245)? Thank you in advance.