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An unbounded operator (or simply operator) T : D(T) → Y is a linear map T from a linear subspace D(T) ⊆ X —the domain of T —to the space Y. [5] Contrary to the usual convention, T may not be defined on the whole space X. An operator T is said to be closed if its graph Γ(T) is a closed set. [6]
Intuitively, the graph of a bounded function stays within a horizontal band, while the graph of an unbounded function does not. In mathematics , a function f {\displaystyle f} defined on some set X {\displaystyle X} with real or complex values is called bounded if the set of its values is bounded .
With suitable modifications, this result can be extended to possibly unbounded operators on infinite-dimensional spaces. Since an everywhere-defined self-adjoint operator is necessarily bounded, one needs to be more attentive to the domain issue in the unbounded case. This is explained below in more detail.
maps every null sequence to a bounded sequence; [3] A null sequence is by definition a sequence that converges to the origin. Thus any linear map that is sequentially continuous at the origin is necessarily a bounded linear map.
Definition A densely defined ... is an example of an unbounded linear operator, since = ... This extension is the Paley–Wiener map. See also
Conversely, a set which is not bounded is called unbounded. The word "bounded" makes no sense in a general topological space without a corresponding metric . Boundary is a distinct concept; for example, a circle (not to be confused with a disk ) in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.
Open mapping theorem for continuous maps [12] [15] — Let : be a continuous linear operator from a complete pseudometrizable TVS onto a Hausdorff TVS . If Im A {\displaystyle \operatorname {Im} A} is nonmeager in Y {\displaystyle Y} then A : X → Y {\displaystyle A:X\to Y} is a (surjective) open map and Y {\displaystyle Y} is a complete ...
The first inequality (that is, ‖ ‖ < for all ) states that the functionals in are pointwise bounded while the second states that they are uniformly bounded. The second supremum always equals ‖ ‖ (,) = ‖ ‖, ‖ ‖ and if is not the trivial vector space (or if the supremum is taken over [,] rather than [,]) then closed unit ball can be replaced with the unit sphere