i have the following problem: i can show that the map $d:B(H)\times B(H)\rightarrow \mathbb{R}$ (with $H$ a separable Hilbert space and $(e_n)_{n\geq1}$) given by:
$$d(x,y)=\sum_{m,n\in\mathbb{N}}{\frac{\left|\left\langle (x-y)e_n,e_m\right\rangle\right|}{2^{n+m}}}$$
where $\langle\cdot,\cdot\rangle$ denotes the inner product on $H$, is a metric for the closed unit ball $A$ of $B(H)$. Now i want to show that this metric on $A$ induces the weak topology. Therefore i have to show that $a_{\iota}\rightarrow a$ (weak) (here $(a_{\iota})$ is a net in $A$) iff $d(a_{\iota},a)\rightarrow 0$. Notice that $a_i\rightarrow a$ weak iff $\langle a_ix,y\rangle\rightarrow\langle ax,y\rangle$ for all $x,y\in H$. The implication "$d(a_i,a)\rightarrow0\Rightarrow a_i\rightarrow a$ (weak)" i can prove, but the other implication seems to be difficult for me because you have to make the right estimations of the sum. Can someone help me with this? Thanky you very much :)
If $a_i\rightarrow a$, weakly then, given $\varepsilon>0$, choose $N$ such that $\sum_{n,m\geq N}^\infty\frac{1}{2^{n+m}}<\varepsilon$. For every $i$, we have $$d(a_i,a)\leq\sum_{n,m=1}^{N-1}\frac{|\langle(a_i-a)(e_n),e_m\rangle|}{2^{n+m}}+\sum_{n,m=N}^{N-1}\frac{2}{2^{n+m}}<\sum_{n,m=1}^{N-1}\frac{|\langle(a_i-a)(e_n),e_m\rangle|}{2^{n+m}}+\varepsilon.$$ Letting $i\rightarrow\infty$ (notice that we have only $(N-1)^2$ terms to cancel), we obtain $\limsup_i d(a_i,a)\leq\varepsilon$, for arbitrary $\varepsilon$, and this means that $a_i\rightarrow a$ in $d$.