- #1

- 4,807

- 32

**[SOLVED] Little bit of convex analysis on a Hilbert space**

## Homework Statement

Let H be a Hilbert space over R and f-->R a function that is bounded below, convex and lower semi continuous (i.e., f(x) [tex]\leq \liminf_{y\rightarrow x}f(y)[/tex] for all x in H).

(a) For all x in H and lambda>0, show that there exists a unique x_lambda in H such that

[tex]\lambda f(x_{\lambda})+||x-x_{\lambda}||^2=\min_{y\in H}(\lambda f(y)+||x-y||^2)[/tex]

## The Attempt at a Solution

Let {y_n} be a sequence such that [tex]\lambda f(y_{n})+||x-y_{n}||^2\rightarrow \inf_{y\in H}(\lambda f(y)+||x-y||^2)[/tex]. If I could show that {y_n} is Cauchy, then by continuity of the norm and lower semi continuity of f, I could conclude that the limit of {y_n} minimizes [tex]\lambda f(y)+||x-y||^2)[/tex].

But do I have enough information to achieve that?

Last edited: