Given that $T: X \to Y$ be a bounded linear operator between Hilbert spaces. Now how to prove that $$||Tx-y||^2+ \alpha ||x||^2$$ is a strictly convex functional where $\alpha > 0$ is a constant and $x \in C$ which is convex and closed subset of $X$.
I know that a function is strictly convex if for $a \neq b$ $$f(\lambda a+(1-\lambda)b) < \lambda f(a)+ (1-\lambda) f(b)$$