‎Jointly ‎continuous of product in $B(H)$ ‎Let ‎$‎B(H)‎$ ‎be the set of‎ ‎bounded ‎operators ‎on a Hilbert space ‎‎$‎‎H$.‎
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I ‎know that ‎$u_{‎\alpha‎}‎\longrightarrow u‎$‎‎‎ ‎in ‎S.O.T ‎if ‎and ‎only ‎if‎  ‎$u_{‎\alpha‎}(x)‎\longrightarrow u(x)‎$ for all‎$‎‎x\in H$.‎
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 I know that 
‎$\cdot : ‎‎B(H)‎\times B(H) ‎\longrightarrow ‎B(H)‎$‎‎‎ ‎such ‎that ‎‎$‎‎(u,v)‎\longmapsto uv‎$‎‎ ‎is ‎separately ‎continuous ‎and ‎jointly ‎continuous ‎on ‎bounded ‎set.
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 Q:I ‎need some ‎nets ‎or ‎sequences ‎in ‎‎$‎B(H)‎$ ‎such ‎that ‎product ‎is ‎not ‎jointly ‎continuous ‎i.e I‎ ‎need ‎‎$(u_{‎\alpha‎}) $‎ ‎and‎$‎ (‎v‎_{‎\alpha‎}‎)$ in $B(H)$ ‎such ‎that ‎ ‎‎$u_{‎\alpha‎}‎\longrightarrow u‎$ and ‎$v_{‎\alpha‎}‎\longrightarrow u‎$ in S.O.T topology but ‎$u_{‎\alpha‎}v_{‎\alpha‎}‎\nrightarrow‎ uv‎$
 A: Let $I$ be the directed set of finite dimensional subspaces of $H$, ordered with inclusion. Define for such a subspace $u_V = 0\cdot \pi_V +\, 2^{|V|}\cdot\pi_{V^\perp}$, here $\pi$ is the projection onto the subspace in the index and $|V|$ the dimension of $V$. This (unbounded) net converges to zero in the strong topology, as $u_V$ restricted to $V$ is zero.
For a general definition of $v_V$ we need the axiom of choice, but for most specific Hilbert space it is probably possible to do this without choice, we'll see that for $\ell^2(\Bbb N)$, and since for most Hilbert spaces you consider you have explicit isomorphisms to $\ell^2(\Bbb N)$ it works without choice for those too.
For any finite dimensional subspace $V$ choose a norm $1$ element $x_V$ from $V^\perp$. Further choose an arbitrary norm one element of $H$, $x$ (this element does not depend on $V$). Let $v_V(x)=\frac{x_V\otimes x^*}{|V|}$, ie the map that sends $x$ to $x_V$ with scaling $\frac1{|V|}$ and zero on the complement. This net converges to $0$ in norm and thus also in the strong operator topology.
Now $\|u_Vv_V(x)\| = \left\|\frac{2^{|V|}}{|V|}x_V\right\|=\frac{2^{|V|}}{|V|}$, so $u_Vv_V$ does not converge to zero in SOT.
To get rid of choice in $\ell^2(\Bbb N)$, let $(e_n)_{n\in\Bbb N}$ be the standard Hilbert basis, let $x_V$ be the first non-zero $\pi_{V^\perp} e_n$ (rescaled to norm one) and $x$ be $e_1$.
