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I want to show that if $X$ is a reflexive Banach space with norm of class $\mathcal{C}^1$ and $f\colon X\to\mathbb{R}\cup \{+\infty\}$ is convex and lower semicontinuous, then $f_{\lambda}$ is differentiable of class $\mathcal{C}^1$.

(where $f_{\lambda}:X\to\mathbb{R}\cup \{+\infty\}$ is the Moreau-Yosida approximation: $$f_\lambda(x)=\inf_{y\in X} \left\{ f(y)+\frac{1}{2\lambda}|x-y|^2\right\})$$

Maybe, this result could be useful: If $g\colon X\to\mathbb{R}$ is convex and differentiable in every point then $g\in\mathcal{C}^1(X)$.

Many thanks in advance.

  • 0
    Do you know if this works when $X$ is Hilbert space? And if so, would the argument generalize to other uniformly convex spaces?2012-01-15
  • 0
    What does $\mathcal{C}^1$ mean in this context?2013-02-07

2 Answers 2