Local Notions
A subset E of a topological vector space X is said to be
- balanced if tE ⊂ E for every scalar |t | ≤ 1
- bounded if for every neighborhood V of 0, then E ⊂ tV when t is sufficiently large.
The definition of boundedness can be weakened a bit; E is bounded if and only if every countable subset of it is bounded. Also, E is bounded if and only if for every balanced neighborhood V of 0, there exists t such that E ⊂ tV. Moreover, when X is locally convex, the boundedness can be characterized by seminorms: the subset E is bounded iff every continuous semi-norm p is bounded on E.
Every topological vector space has a local base of absorbing and balanced sets.
A sequence {xn} is said to be Cauchy if for every neighborhood V of 0, the difference xm − xn belongs to V when m and n are sufficiently large. Every Cauchy sequence is bounded, although Cauchy nets or Cauchy filters may not be bounded. A topological vector space where every Cauchy sequence converges is sequentially complete but may not be complete (in the sense Cauchy filters converge). Every compact set is bounded.
Read more about this topic: Topological Vector Space
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