Exercise 11, § 1

Let E be a Hilbert space, $(x_n)$ a sequence of points of E which converges weakly to a point $a$. For every $y \in E$, put

Now prove that one has $d(y)^2 = d(a)^2 + \|y - a\|^2$ and $D(y)^2 = D(a)^2 + \|y - a\|^2$. If $\alpha$ and $\beta$ are two real numbers such that $0 \leq \alpha \leq \beta$, give examples where $d(a) = \alpha$ and $D(a) = \beta$.