Open mapping theorem — Let : be a surjective linear map from an complete pseudometrizable TVS onto a TVS and suppose that at least one of the following two conditions is satisfied: . is a Baire space, or; is locally convex and is a barrelled space,; If is a closed linear operator then is an open mapping. If is a continuous linear operator and is Hausdorff then is (a closed linear …

The uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field.In its basic form, it asserts that for a family of continuous linear operators (and thus bounded operators) whose domain is a Banach …

The closed graph theorem is an important result in functional analysis that guarantees that a closed linear operator is continuous under certain conditions. ... Discontinuous linear map; Kakutani fixed-point theorem; Open mapping theorem (functional analysis) – Condition for a linear operator to be open; Ursescu theorem – Generalization of ...

Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer.It states that for any continuous function mapping a compact convex set to itself there is a point such that () =.The simplest forms of Brouwer's theorem are for continuous functions from a closed interval in the real numbers to itself or from a closed disk to itself.

Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. Definition. A Banach space is a complete normed space (, ‖ ‖). A ... Further, by the open mapping theorem, if there is a bounded linear operator from the Banach space onto the Banach space , then is ...

The open mapping theorem states that a continuous surjective linear transformation from one Banach space to another is an open mapping meaning that it sends ... Rigged Hilbert space – Construction linking the study of "bound" and continuous eigenvalues in functional analysis; Topological vector space – Vector space with a notion of ...

Manifolds need not be connected (all in "one piece"); an example is a pair of separate circles.. Manifolds need not be closed; thus a line segment without its end points is a manifold.They are never countable, unless the dimension of the manifold is 0.Putting these freedoms together, other examples of manifolds are a parabola, a hyperbola, and the locus of points on a cubic …

The following statement appears in Hamilton (1982): . Let F and G be tame Fréchet spaces, let be an open subset, and let : be a smooth tame map. Suppose that for each the linearization : is invertible, and the family of inverses, as a map , is smooth tame. Then P is locally invertible, and each local inverse is a smooth tame map.. Similarly, if each linearization is only injective, and a ...

A multi-index of size is an element in (given that is fixed, if the size of multi-indices is omitted then the size should be assumed to be ).The length of a multi-index = (, …,) is defined as + + and denoted by | |. Multi-indices are particularly useful when dealing with functions of several variables, in particular, we introduce the following notations for a given multi-index = (, …,):

The Baire category theorem (BCT) is an important result in general topology and functional analysis.The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that the intersection of countably many dense open sets is still dense). It is used in the proof of results in many areas of analysis and …