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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.

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    Perhaps you need to explain what is C1?2012-01-23

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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 $f(y) \geq \ell(y) + r$.

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    Ahmm, many thanks2012-01-25