For a seminar I am working on a Moreau-Yosida regularization in Banach spaces.

The regularization is defined by

$$f_\lambda(x) := \inf \left \{ \frac{\|x-y\|^2}{2\lambda} +f(y) : y \in X \right \}, ~ x \in X$$

where $X$ is a reflexive and strictly convex Banach space, $f: X \rightarrow \mathbb{R} \cup \{+\infty\}$ lower-semicontinuous, proper and convex, $\lambda > 0$.

In Convexity and Optimization in Banach Spaces, V. Barbu and T. Precupanu, Springer, 2012, the authors prove that the infimum defining $f_\lambda(x)$ is attained for all $x \in X$.

From that they deduce that $f_\lambda$ is convex and lower-semicontinious, but, unfortunately, they don't elaborate the argument. However, I do not see how it follows from what is given.

As far as I am aware, most literature deals with the case where $X$ is Hilbert, I could not finde the result stated above for the general case in another publication.

Question: Maybe someone can recommed me a paper where this result is proven? Or is there an obvious argument which I am missing at the moment?

Thank you in advance