= Bounded set in a topological vector space
= Bounded sets in a topological vector space
{synonym}
A subset $B$ of a <topological vector space> is bounded when every neighborhood $U$ of zero absorbs it: $B\subseteq tU$ for all sufficiently large positive $t$. In a <locally convex space> defined by <seminorms> $p_\lambda$, this is equivalent to $\sup_{x\in B}p_\lambda(x)<\infty$ for every $\lambda$. In particular, a bounded subset of the <Schwartz space> has a uniform bound for every Schwartz seminorm; it need not have uniformly bounded supports.
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