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
Famous quotes containing the words local and/or notions:
“This is the only “wet” community in a wide area, and is the rendezvous of cow hands seeking to break the monotony of chuck wagon food and range life. Friday night is the “big time” for local cowboys, and consequently the calaboose is called the “Friday night jail.””
—Administration in the State of Texa, U.S. public relief program (1935-1943)
“What is termed Sin is an essential element of progress. Without it the world would stagnate, or grow old, or become colourless. By its curiosity Sin increases the experience of the race. Through its intensified assertion of individualism it saves us from monotony of type. In its rejection of the current notions about morality, it is one with the higher ethics.”
—Oscar Wilde (1854–1900)