insight into the spaces themselves. The article operator topologies discusses topologies on spaces of linear maps between normed spaces, whereas this article...

weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a...

field of functional analysis there are several standard topologies which are given to the algebra B(X) of bounded linear operators on a Banach space X. Let...

of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties...

topological vector space – Vector space with a topology defined by convex open sets Positive linear functional Topologies on spaces of linear maps Unbounded operator –...

order topology makes X into a completely normal Hausdorff space. The standard topologies on R, Q, Z, and N are the order topologies. If Y is a subset of X...

composition of linear maps. If X {\displaystyle X} and Y {\displaystyle Y} are normed spaces, they are isomorphic normed spaces if there exists a linear bijection...

continuous linear transformations, including topologies on the vector spaces in the above, and many of the major examples are function spaces carrying a...

all linear maps φ : V → F {\displaystyle \varphi :V\to F} (linear functionals). Since linear maps are vector space homomorphisms, the dual space may be...

metric spaces and between topological spaces. The latter are the most general continuous functions, and their definition is the basis of topology. A stronger...

a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments...

convex topologies on the vector spaces of a pairing. A pairing is a triple ( X , Y , b ) {\displaystyle (X,Y,b)} consisting of two vector spaces over a...

spaces should match the topology. For example, instead of considering all linear maps (also called functionals) V → W , {\displaystyle V\to W,} maps between...

topology – Dual space topology of uniform convergence on some sub-collection of bounded subsets Strong topology Topologies on spaces of linear maps Weak...

function on its vector space. All linear maps between finite-dimensional vector spaces are also continuous. An isometry between two normed vector spaces is...

properties of such objects that are common to all vector spaces. Linear maps are mappings between vector spaces that preserve the vector-space structure...

unique. There exist numerous topologies on any given finite set. Such spaces are called finite topological spaces. Finite spaces are sometimes used to provide...

compactness, were developed in general metric spaces. In general topological spaces, however, these notions of compactness are not necessarily equivalent...

index) – Annotated index of various modes of convergence Net (mathematics) – Generalization of a sequence of points Topologies on spaces of linear maps...

Sobolev spaces. Many topological vector spaces are spaces of functions, or linear operators acting on topological vector spaces, and the topology is often...

{\displaystyle K} represent linear maps of vector spaces, say ⁠ K n → K n {\displaystyle K^{n}\to K^{n}} ⁠, and thus linear maps ψ : P n − 1 → P n − 1 {\displaystyle...

In mathematics, linear maps form an important class of "simple" functions which preserve the algebraic structure of linear spaces and are often used as...

of bounded subsets Reflexive space – Locally convex topological vector space Semi-reflexive space Strong topology Topologies on spaces of linear maps...

set' is called a topology. A set with a topology is called a topological space. Metric spaces are an important class of topological spaces where a real,...

index of various modes of convergence Strong operator topology – Locally convex topology on function spaces Topologies on spaces of linear maps Weak topology –...

preorder on the space. Spaces with an Alexandrov topology are also known as Alexandrov-discrete spaces or finitely generated spaces. The latter name...

nor σ {\displaystyle \sigma } -quasi-barrelled. Mackey topology Topologies on spaces of linear maps Bourbaki 1987, p. IV.4. Grothendieck 1973, p. 107. Schaefer...

found in the article on spaces of test functions and distributions and the articles on polar topologies and dual systems. A linear map from D ′ ( U ) {\displaystyle...

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...

bounded family of continuous linear operators between Banach spaces is equicontinuous. Let X and Y be two metric spaces, and F a family of functions from...