Let $X$ be a Banach space, $Y$ be a norm linear space, $(A_n)\in \Bbb B(X,Y)$ be such that for every $x\in X, ||(A_n-A)x||\rightarrow0$ for some $A\in \Bbb B(X,Y)$. If $K:X\rightarrow Y$ is a compact operator, then prove that $||(A_n-A)K||\rightarrow 0$
Prove the following corollary.
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functional-analysis
operator-theory
compact-operators
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2Pointwise convergence implies uniform convergence on compact sets for continuous maps. – 2017-02-04
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0@Daniel Fischer can you elaborate more? How to apply this clue? – 2017-02-04
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0First correct the condition on $K$, the codomain of $K$ must be $X$ for the composition to be defined. Then recall the definition of a compact operator. (There are several equivalent definitions, the one using $T(B)$ where $B$ is the unit ball of the domain is the useful one.) – 2017-02-04