Taking the supremum over in the unit ball of and over it follows that
There are also simple proofs not using the Baire theorem (Sokal 2011).
Corollary—If a sequence of bounded operators converges pointwise, that is, the limit of exists for all then these pointwise limits define a bounded linear operator
The above corollary does not claim that converges to in operator norm, that is, uniformly on bounded sets. However, since is bounded in operator norm, and the limit operator is continuous, a standard "" estimate shows that converges to uniformly on compact sets.
Corollary—Any weakly bounded subset in a normed space is bounded.
Indeed, the elements of define a pointwise bounded family of continuous linear forms on the Banach space which is the continuous dual space of By the uniform boundedness principle, the norms of elements of as functionals on that is, norms in the second dual are bounded. But for every the norm in the second dual coincides with the norm in by a consequence of the Hahn–Banach theorem.
Let denote the continuous operators from to endowed with the operator norm. If the collection is unbounded in then the uniform boundedness principle implies:
In fact, is dense in The complement of in is the countable union of closed sets By the argument used in proving the theorem, each is nowhere dense, i.e. the subset is of first category. Therefore is the complement of a subset of first category in a Baire space. By definition of a Baire space, such sets (called residual sets) are dense. Such reasoning leads to the principle of condensation of singularities, which can be formulated as follows:
Theorem—Let be a Banach space, a sequence of normed vector spaces, and an unbounded family in Then the set
is a residual set, and thus dense in
The complement of is the countable union
of sets of first category. Therefore, its residual set is dense.
Example: pointwise convergence of Fourier series
Let be the circle, and let be the Banach space of continuous functions on with the uniform norm. Using the uniform boundedness principle, one can show that there exists an element in for which the Fourier series does not converge pointwise.
where is the -th Dirichlet kernel. Fix and consider the convergence of The functional defined by
is bounded. The norm of in the dual of is the norm of the signed measure namely
It can be verified that
So the collection is unbounded in the dual of Therefore, by the uniform boundedness principle, for any the set of continuous functions whose Fourier series diverges at is dense in
More can be concluded by applying the principle of condensation of singularities. Let be a dense sequence in Define in the similar way as above. The principle of condensation of singularities then says that the set of continuous functions whose Fourier series diverges at each is dense in (however, the Fourier series of a continuous function converges to for almost every by Carleson's theorem).
Attempts to find classes of locally convex topological vector spaces on which the uniform boundedness principle holds eventually led to barrelled spaces. That is, the least restrictive setting for the uniform boundedness principle is a barrelled space, where the following generalized version of the theorem holds (Bourbaki 1987, Theorem III.2.1) harv error: no target: CITEREFBourbaki1987 (help):
is a bounded subset of ; if is a normed space then this happens if and only if there exists some real such that
The below theorem's conclusion that the set is necessarily equal to all to can be deduced, with the help of the first part of the following theorem, from the equicontinuity of and the fact that every single subset of is also a bounded subset.
Let be balanced neighborhoods of the origin in satisfying It must be shown that there exists a neighborhood of the origin in such that for every Let
which is a closed subset of (because it is an intersection of closed subsets) that for every also satisfies and
(as will be shown, the set is in fact a neighborhood of the origin in because the topological interior of in is non-empty). If then being bounded in implies that there exists some integer such that so if then Since was arbitrary,
This proves that
Because is of the second category in the same must be true of at least one of the sets for some The map defined by is a (surjective) homeomorphism, so the set is necessarily of the second category in Because is closed and of the second category in its topological interior in is non-empty. Pick The map defined by being a homeomorphism implies that
is a neighborhood of in (which implies that the same is true of the superset ). And so for every
This proves that is equicontinuous.
Proof that :
Because is equicontinuous, if is bounded in then is uniformly bounded in In particular, for any because is a bounded subset of is a uniformly bounded subset of Thus
Barrelled space– A topological vector space with near minimum requirements for the Banach–Steinhaus theorem to hold.
Ursescu theorem– Theorem that simultaneously generalizes the closed graph, open mapping, and Banach–Steinhaus theorems
↑ Let be a neighborhood of the origin in Since is equicontinuous, there exists a neighborhood of the origin in such that for every Because is bounded in there exists some real such that if then So for every and every which implies that Thus is bounded in
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.
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. A topological vector space is a vector space which is also a topological space, this implies that vector space operations be continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.
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 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 is reflexive if and only if the canonical evaluation map from into its bidual is surjective; in this case the normed space is necessarily also a Banach space. In 1951, R. C. James discovered a Banach space, now known as James' space, that is not reflexive but is nevertheless isometrically isomorphic to its bidual.
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 functional analysis, a bounded linear operator is a linear transformation between topological vector spaces (TVSs) and that maps bounded subsets of to bounded subsets of If and are normed vector spaces, then is bounded if and only if there exists some such that for all in
In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus sequences of functions.
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.
The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded interval has a uniformly convergent subsequence. The main condition is the equicontinuity of the family of functions. The theorem is the basis of many proofs in mathematics, including that of the Peano existence theorem in the theory of ordinary differential equations, Montel's theorem in complex analysis, and the Peter–Weyl theorem in harmonic analysis and various results concerning compactness of integral operators.
In linear algebra, the transpose of a linear map between two vector spaces, defined over the same field, is an induced map between the dual spaces of the two vector spaces. The transpose or algebraic adjoint of a linear map is often used to study the original linear map. This concept is generalised by adjoint functors.
In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed "size"
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 functional and convex analysis, and related disciplines of mathematics, the polar set is a special convex set associated to any subset of a vector space lying in the dual space The bipolar of a subset is the polar of but lies in .
In functional analysis and related areas of mathematics a polar topology, topology of -convergence or topology of uniform convergence on the sets of is a method to define locally convex topologies on the vector spaces of a pairing.
In mathematics, a nuclear space is a topological vector space that can be viewed as a generalization of finite dimensional Euclidean spaces that is different from Hilbert spaces. Nuclear spaces have many of the desirable properties of finite-dimensional vector spaces. The topology on them can be defined by a family of seminorms whose unit balls decrease rapidly in size. Vector spaces whose elements are "smooth" in some sense tend to be nuclear spaces; a typical example of a nuclear space is the set of smooth functions on a compact manifold.
In mathematics, a linear map is a mapping between two modules that preserves the operations of addition and scalar multiplication.
The strongest locally convex topological vector space (TVS) topology on , the tensor product of two locally convex TVSs, making the canonical map continuous is called the projective topology or the π-topology. When X ⊗ Y is endowed with this topology then it is denoted by and called the projective tensor product of X and Y.
In mathematics, the injective tensor product of two topological vector spaces (TVSs) was introduced by Alexander Grothendieck and was used by him to define nuclear spaces. An injective tensor product is in general not necessarily complete, so its completion is called the completed injective tensor products. Injective tensor products have applications outside of nuclear spaces. In particular, as described below, up to TVS-isomorphism, many TVSs that are defined for real or complex valued functions, for instance, the Schwartz space or the space of continuously differentiable functions, can be immediately extended to functions valued in a Hausdorff locally convex TVS Y without any need to extend definitions from real/complex-valued functions to Y-valued functions.
In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector spaces (TVSs).
In functional analysis, every C*-algebra is isomorphic to a subalgebra of the C*-algebra of bounded linear operators on some Hilbert space H. This article describes the spectral theory of closed normal subalgebras of
This is a glossary for the terminology in a mathematical field of functional analysis.
This page is based on this Wikipedia article Text is available under the CC BY-SA 4.0 license; additional terms may apply. Images, videos and audio are available under their respective licenses.