In mathematics, the bounded inverse theorem (or inverse mapping theorem) is a result in the theory of bounded linear operators on Banach spaces. It states that a bijective bounded linear operator T from one Banach space to another has bounded inverse T−1. It is equivalent to both the open mapping theorem and the closed graph theorem.
This theorem may not hold for normed spaces that are not complete. For example, consider the space X of sequences x : N → R with only finitely many non-zero terms equipped with the supremum norm. The map T : X → X defined by
is bounded, linear and invertible, but T−1 is unbounded. This does not contradict the bounded inverse theorem since X is not complete, and thus is not a Banach space. To see that it's not complete, consider the sequence of sequences x(n) ∈ X given by
converges as n → ∞ to the sequence x(∞) given by
which has all its terms non-zero, and so does not lie in X.
The completion of X is the space of all sequences that converge to zero, which is a (closed) subspace of the ℓp space ℓ∞(N), which is the space of all bounded sequences. However, in this case, the map T is not onto, and thus not a bijection. To see this, one need simply note that the sequence
is an element of , but is not in the range of .
In mathematics, more specifically in functional analysis, a Banach space is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.
The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space "interesting". Another version of the Hahn–Banach theorem is known as the Hahn–Banach separation theorem or the hyperplane separation theorem, and has numerous uses in convex geometry.
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis.
In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) such that the canonical evaluation map from X into its bidual is an isomorphism of TVSs. Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space X is reflexive if and only if the canonical evaluation map from X into its bidual is surjective; in this case the normed space is necessarily also a Banach space. Note that in 1951, R. C. James discovered a non-reflexive Banach space that is isometrically isomorphic to its bidual.
In linear algebra, a linear form is a linear map from a vector space to its field of scalars. If vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the matrix product with the row vector on the left and the column vector on the right. In general, if V is a vector space over a field k, then a linear functional f is a function from V to k that is linear:
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces. All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically not Banach spaces.
In mathematics, an isometry is a distance-preserving transformation between metric spaces, usually assumed to be bijective.
In functional analysis, a bounded linear operator is a linear transformation L : X → Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector spaces, then L is bounded if and only if there exists some M ≥ 0 such that for all x in X,
In mathematics, the closed graph theorem is a basic result which characterizes continuous functions in terms of their graphs. In particular, they give conditions when functions with closed graphs are necessarily continuous. In mathematics, there are several results known as the "closed graph theorem".
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.
In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proof identifies the unit ball with the weak-* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact.
In functional analysis and related areas of mathematics, a barrelled space is a topological vector spaces (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector. A barrelled set or a barrel in a topological vector space is a set that is convex, balanced, absorbing, and closed. Barrelled spaces are studied because a form of the Banach–Steinhaus theorem still holds for them.
In mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that a topological space possesses the minimum amount of structure needed to address questions of continuity. Bornological spaces are distinguished by that property that a linear maps from a bornological space into any locally convex spaces is continuous if and only if it a bounded linear operator.
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem, is a fundamental result which states that if a continuous linear operator between Banach spaces is surjective then it is an open map.
In mathematics, particularly in functional analysis, a webbed space is a topological vector space designed with the goal of allowing the results of the open mapping theorem and the closed graph theorem to hold for a wider class of linear maps. A space is called webbed if there exists a collection of sets, called a web that satisfies certain properties. Webs were first investigated by de Wilde.
In the branch of mathematics called functional analysis, when a topological vector space X admits a direct sum decomposition X ≅Y ⊕ Z, the spaces Y and Z are called complements of each other. This happens if and only if the addition map Y × Z → X, which is defined by (y, z) ↦ y + z, is a homeomorphism. Note that while this addition map is always continuous, it may fail to be a homeomorphism, which is why this definition is needed.
In functional analysis and related areas of mathematics, an almost open linear map between topological vector spacess (TVSs) is a linear operator that satisfies a condition similar to, but weaker than, the condition of being an open map.
In functional analysis and related areas of mathematics, a metrizable topological vector spaces (TVS) is a TVS whose topology is induced by a metric. An LM-space is an inductive limit of a sequence of locally convex metrizable TVS.
F. Riesz's theorem is an important theorem in functional analysis that states that a Hausdorff topological vector space (TVS) is finite-dimensional if and only if it is locally compact. The theorem and its consequences are used ubiquitously in functional analysis, often used without being explicitly mentioned.
In mathematics, particularly in functional analysis and topology, the closed graph theorem is a fundamental result stating that a linear operator with a closed graph will, under certain conditions, be continuous. The original result has been generalized many times so there are now many theorems referred to as "closed graph theorems."