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A topological vector space $X$ is a vector space over a topological field $K$ (most often the real or complex numbers with their standard topologies) that is endowed with a topology such that vector addition $X \times X \to X$ and scalar multiplication $K \times X \to X$ are continuous functions (where the domains of these functions are endowed with product topologies). Now I want to show that in a topological vector space, the only bounded subspace is $\{0\}$.

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    $\begingroup$ What definition of bounded subspace used? [your question didn't specify a metric on $X$] $\endgroup$ – coffeemath Apr 8 '18 at 3:52

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