mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between...

called the operator norm of L {\displaystyle L} and denoted by ‖ L ‖ . {\displaystyle \|L\|.} A linear operator between normed spaces is continuous if and...

In functional analysis, a branch of mathematics, a compact operator is a linear operator T : X → Y {\displaystyle T:X\to Y} , where X , Y {\displaystyle...

may be continuous. If its domain and codomain are the same, it will then be a continuous linear operator. A linear operator on a normed linear space is...

invariant continuous linear operator on L1 is the convolution with a finite Borel measure. More generally, every continuous translation invariant continuous linear...

{\displaystyle f} is a bounded linear operator and so is continuous. In fact, to see this, simply note that f is linear, and therefore ‖ f ( x ) − f (...

Theorems connecting continuity to closure of graphs Continuous linear operator Densely defined operator – Function that is defined almost everywhere (mathematics)...

topological vector spaces (TVSs) X and Y. An integral linear operator is a continuous linear operator that arises in a canonical way from an integral bilinear...

mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its operator norm. Formally, it...

functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication...

Furthermore, the kernel of a continuous projection (in fact, a continuous linear operator in general) is closed. Thus a continuous projection P {\displaystyle...

functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a generalisation of the set of eigenvalues...

Schauder), is a fundamental result that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map. A...

are the continuous linear operators defined on Banach and Hilbert spaces. These lead naturally to the definition of C*-algebras and other operator algebras...

specifically in operator theory, each linear operator A {\displaystyle A} on an inner product space defines a Hermitian adjoint (or adjoint) operator A ∗ {\displaystyle...

functional analysis, a normal operator on a complex Hilbert space H {\displaystyle H} is a continuous linear operator N : H → H {\displaystyle N\colon...

x} , which is a continuous linear operator of rank 1 and thus a Hilbert–Schmidt operator; moreover, for any bounded linear operator A {\displaystyle...

function. In a topological sense, it is a linear operator that is defined "almost everywhere". Densely defined operators often arise in functional analysis as...

other examples) The most basic operators are linear maps, which act on vector spaces. Linear operators refer to linear maps whose domain and range are...

bounded linear operators on a Banach space X. Let ( T n ) n ∈ N {\displaystyle (T_{n})_{n\in \mathbb {N} }} be a sequence of linear operators on the Banach...

Every continuous linear operator is a bounded linear operator and if dealing only with normed spaces then the converse is also true. That is, a linear operator...

analysis, a branch of mathematics, a closed linear operator or often a closed operator is a linear operator whose graph is closed (see closed graph property)...

continuous linear operators from X {\displaystyle X} into Y {\displaystyle Y} . Suppose that F {\displaystyle F} is a collection of continuous linear...

Thus, a linear operator A is the infinitesimal generator of a uniformly continuous semigroup if and only if A is a bounded linear operator. If X is a...

adjoint. A particular case is that of a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties: A...

Y} is a weakly continuous linear operator between topological vector spaces X {\displaystyle X} and Y {\displaystyle Y} with continuous dual spaces X ′...

mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may...

boundedness principle states that a pointwise bounded family of continuous linear operators between Banach spaces is equicontinuous. Let X and Y be two metric...

mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite...

is a subspace of the dual space, corresponding to continuous linear functionals, called the continuous dual space. Dual vector spaces find application in...