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ordered topological vector space∗
2013-03-21 22:47:44
Let k be either R or C considered as a field. An ordered topological vector
space L, (ordered t.v.s for short) is
• a topological vector space over k, and
• an ordered vector space over k, such that
• the positive cone L+ of L is a closed subset of L.
The last statement can be interpreted as follows: if a sequence of nonnegative elements xi of L converges to an element x, then x is non-negative.
Remark. Let L, M be two ordered t.v.s., and f : L → M a linear transformation that is monotone. Then if 0 ≤ x ∈ L, 0 ≤ f (x) ∈ M also. Therefore
f (L+ ) ⊆ M + . Conversely, a linear map that is invariant under positive cones
is monotone.
∗ hOrderedTopologicalVectorSpacei created: h2013-03-21i by: hCWooi version: h39347i
Privacy setting: h1i hDefinitioni h06F20i h46A40i h06F30i
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