I have some difficulty to understand an argument in a proof about $\alpha$ convex functions.
Consider $J : K\to \mathbb{R}$ where $K\subset V$ is a convex and closed set, $V$ an Hilbert space and $J$ is $\alpha$ convex and continuous.
The property is the following
There exists $\eta\in\mathbb{R}$ and $\gamma>0$ such that
$$ J(v)\geq \gamma\lVert v\rVert^{2} + \eta $$
The proof is as follows : first, we take for granted that there exists $L\in V^{*}$ and $\delta\in\mathbb{R}$ such that
$$ J(v)\geq L(v) + \delta\quad\forall v\in K $$
(Can be proven with the separation theorem)
Fix $y\in K$. Take an arbitrary $v\in K$, by the $\alpha$ convexity of $J$ we have
$$ J\left(\frac{y+v}{2}\right)\leq \frac{J(y)}{2} + \frac{J(v)}{2} - \frac{\alpha}{8}\lVert v - y\rVert^{2} $$
It follows, using the first remark and the Cauchy Schwarz inequality :
$$ J(v)\geq 2J(\frac{y+v}{2}) + J(y) +\frac{\alpha}{4}\left(\lVert x\rVert^{2} - 2\lVert x\rVert\lVert y\rVert + \lVert y\rVert^{2}\right)\geq L(y) + L(v) + + 2\delta +\frac{\alpha}{4}\left(\lVert x\rVert^{2} - 2\lVert x\rVert\lVert y\rVert + \lVert y\rVert^{2}\right) $$
Now using the operator norm we have $\lvert L(v)\rvert\leq\lVert L\rVert_{V^{*}}\lVert v\rVert$. Thus by adding the negative quantity $\lvert L(v)\rvert -\lVert L\rVert_{V^{*}}\lVert v\rVert$ we get
$$ J(v)\geq L(y) + L(v) + 2\delta +\frac{\alpha}{4}\lVert x\rVert^{2} - \frac{\alpha}{2}\lVert x\rVert\lVert y\rVert +\frac{\alpha}{4}\lVert y\rVert^{2} \geq\frac{\alpha}{4}\lVert x\rVert^{2} -\left(\frac{\alpha}{2}\lVert y\rVert + \lVert L\rVert_{V^{*}}\right)\lVert v\rVert + C $$
Where $C = L(y) + 2\delta +\frac{\alpha}{4}\lVert y\rVert^{2}$
Then the author conclude by saying that, if we choose the good $\eta$ we get
$$ J(v)\geq \frac{\alpha}{4}\lVert x\rVert^{2} +\eta $$
And this is this last step that I do not understand since in the following expression $-\left(\frac{\alpha}{2}\lVert y\rVert + \lVert L\rVert_{V^{*}}\right)\lVert v\rVert + C$ there is still a dependence on $v$.
At best we can get this expression
$$ J(v)\geq\lVert v\rVert^{2}\left(\frac{\alpha}{4}-\left(\lVert L\rVert_{V^*} +\frac{\alpha}{2}\right)\frac{1}{\lVert v\rVert}\right) + C $$
But then we cannot lower bound this as far as I know.
Have you an idea on what this argument should be true( if it is) please ?
Thank you a lot