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Let $H$ be a Hilbert space and assume we have three converging sequences: $u_n\rightarrow u$ in $H$, $v_n\rightarrow v$ in $H$ and $\lambda_n\rightarrow \lambda$ in $\mathbb{C}$. I would like to prove that the (bounded) operator $\lambda_n(u_n,.)v_n$ converges in operator norm to $\lambda(u,.)v$.

I thought of first trying to show $||\lambda_n(u_n,x)v_n-\lambda(u,x)v||\rightarrow 0$ for every $x$, using tricks like $u_n=u_n-u+u$, and then deduce the operator norm convergence, but neither result is obvious to me.

Please help :-)

Pete
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1 Answers1

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I will use the notation $x\bigcirc y$ for operator $(y,\cdot)x$

Let $w_n=\lambda_n u_n$, then $w_n\to w:=\lambda u$. Now it is remains to show that $ v_n\bigcirc w_n \to v\bigcirc w$. It is easy to do using trick you mentioned before becasue $$ \Vert x\bigcirc y\Vert\leq\Vert x\Vert\Vert y\Vert $$ (in fact we have an equality here)

Norbert
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