Topological Vector Space - Local Notions

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.

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