Show that $f=1/2 ||T(x)||^2$ is Gateaux differentiable everywhere.

Question

Let $T$ be a bounded linear transformation from a real Hilbert space $H$ onto itself and define $f$ by $$f(x)=\dfrac{1}{2}||T(x)||^2.$$ How can I show that $f$ is Gateaux differentiable everywhere?

Thanks!

Answer 1

To show that $f$ is Gateaux differentiable, you must show that for every $x,y\in H$ the derivative $$ \left.\frac{d}{dt} f(x + ty)\right|_{t=0}$$ exists. So let's compute $f(x + ty)$: $$f(x+ty) = \frac{1}{2}\|T(x + ty)\|^2 = \frac{1}{2}\langle T(x + ty), T(x + ty)\rangle = \frac{1}{2}\left[\|T(x)\|^2 + 2t\langle T(x), T(y)\rangle + t^2\|T(y)\|^2\right]$$ Notice that since we have fixed $x$ and $y$ here, this is just a quadratic function of $t\in \mathbb{R}$. It is in particular differentiable, with $$\left.\frac{d}{dt}f(x + ty)\right|_{t=0} = \langle T(x), T(y)\rangle$$