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Found problems: 126

1986 Miklós Schweitzer, 6
Tags: topology , vector
Let $U$ denote the set $\{ f\in C[0, 1] \colon |f(x)|\leq 1\, \mathrm{for}\,\mathrm{all}\, x\in [0, 1]\}$. Prove that there is no topology on $C[0, 1]$ that, together with the linear structure of $C[0,1]$, makes $C[0,1]$ into a topological vector space in which the set $U$ is compact. (Assume that topological vector spaces are Hausdorff) [V. Totik]