Little bit of convex analysis on a Hilbert space

[quasar987]

[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) $$\leq \liminf_{y\rightarrow x}f(y)$$ 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

$$\lambda f(x_{\lambda})+||x-x_{\lambda}||^2=\min_{y\in H}(\lambda f(y)+||x-y||^2)$$

The Attempt at a Solution

Let {y_n} be a sequence such that $$\lambda f(y_{n})+||x-y_{n}||^2\rightarrow \inf_{y\in H}(\lambda f(y)+||x-y||^2)$$. 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 $$\lambda f(y)+||x-y||^2)$$.

But do I have enough information to achieve that?

 

[quasar987]

What other methods are there to show a function attains it's minimum?

I know a continuous function on a compact set attains its min. But this is not useful here.

I know the distance of a point to a closed convex set in a hilbert space is minimized by a point of the convex. But this is not useful here either as far as i can see.

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