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Let $A$ and $B$ be two Banach spaces and consider the linear space $\mathbf{Ban}(A, B)$ of bounded linear maps with the topology of uniform convergence on compact sets. Since this topology is generated by the family of seminorms $p_K = \sup\{\|Ta|\ \colon a\in K\}$ for $K$ a compact subset of $A$, the topology is Hausdorff and locally convex. My question is: is it complete? And if this is known, any quotable reference?

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  • $\begingroup$ What have you tried? What happens if you use the defintiion of complete for a TVS? $\endgroup$
    – supinf
    Oct 26, 2020 at 10:08

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This indeed seems to be easier than I (did not) thought of as supinf intimates. Sketch of proof: a Cauchy net in the topology of uniform convergence on compacta is pointwise Cauchy so it has a pointwise limit. It is easy to see that the pointwise limit is a linear map, and by taking pointwise limits in the Cauchy condition, convergence to this limit is uniform in compacta. To see that the limit is bounded it suffices to prove it preserves null sequences, but a null sequence is a relatively compact set so the desired conclusion follows from an interchange of limits which is possible because convergence is uniform on the compact set constituted by the sequence and its $0$ limit.

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  • $\begingroup$ I will mark it as solved as soon as I am satisfied that I have not botched the proof. $\endgroup$ Oct 26, 2020 at 13:24

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