A question about the coercivity of a lsc and convex function.

Question

I was doing a proof and I need to show a result to conclude it:

$X$ is a reflexive Banach space with a norm, $\|\cdot\|$, of class $\mathcal{C}^1$.

$f:X\to\overline{\mathbb{R}}$ is lower semicontinuous and convex.

If $\lambda>0$ and $x\in X$ are prefixed, I define $\Phi:X\to\overline{\mathbb{R}}$ such that $\Phi(y)=f(y)+\frac{1}{2\lambda}\|x-y\|^2$.

Then I need to show that $\Phi$ is coercive, this is: $\displaystyle\lim_{\|y\|\to\infty} \Phi(y)=+\infty$.

Maybe the result isn't true... Thanks.

Answer 1

Hint 1: The triangle inequality implies that $\|y\| - \|x\| \leq \|y-x\|$.

Hint 2: Convexity implies that there exists a linear functional $\ell:X\to\mathbb{R}$ and a real number $r$ such that $f(y) \geq \ell(y) + r$.