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Miklós Schweitzer 1986, Problem 6

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September 12, 2016
Miklos Schweitzercollege conteststopologyvector

Problem Statement

Let UU denote the set {f∈C[0,1] ⁣:∣f(x)∣≤1 for all x∈[0,1]}\{ f\in C[0, 1] \colon |f(x)|\leq 1\, \mathrm{for}\,\mathrm{all}\, x\in [0, 1]\}. Prove that there is no topology on C[0,1]C[0, 1] that, together with the linear structure of C[0,1]C[0,1], makes C[0,1]C[0,1] into a topological vector space in which the set UU is compact. (Assume that topological vector spaces are Hausdorff) [V. Totik]
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