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.