I am trying to show that every Hilbert space $H$ is strictly smooth with modulus of smoothness $\phi_H(t)=\sqrt{1+t^2} -1 $.
To show this I think I should show $H$ is uniformly smooth first. $\|\frac{h+tg}{2}\|+\|\frac{h-tg}{2}\|=1+o(t)$ uniformly for $\|h\|=1$, $\|g\|\le 1$. Modulus of smoothness is $\phi_H(t)= \sup_{\|h\|=1, \|g\|\le 1} \|\frac{h+tg}{2}\|+\|\frac{h-tg}{2}\| -1$ And then I don't know how to show $H$ is strictly smooth.