continuous map is bounded, however the converse fails; a bounded operator need not be continuous. This also means that boundedness is no longer equivalent...

norm defined on the space of bounded linear operators between two given normed vector spaces. Informally, the operator norm ‖ T ‖ {\displaystyle \|T\|}...

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

Every sequentially continuous linear operator is bounded. Function bounded on a neighborhood and local boundedness In contrast, a map F : X → Y {\displaystyle...

image of any bounded subset of X {\displaystyle X} under T {\displaystyle T} is totally bounded in Y {\displaystyle Y} . If a linear operator is compact...

In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator A : H → H {\displaystyle A\colon H\to...

transpose). The above definition of an adjoint operator extends verbatim to bounded linear operators on Hilbert spaces H {\displaystyle H} . The definition...

In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Non-trivial examples...

{\displaystyle X\times Y} ). A bounded operator is a closed operator. Here are examples of closed operators that are not bounded. If ( X , τ ) {\displaystyle...

term "unbounded operator" can be misleading, since "unbounded" should sometimes be understood as "not necessarily bounded"; "operator" should be understood...

characteristics, such as bounded linear operators or closed operators, and consideration may be given to nonlinear operators. The study, which depends...

is bounded. (However, a continuous function must be bounded if its domain is both closed and bounded.) Bounded set Compact support Local boundedness Uniform...

\operatorname {Dom} (A)=H} then A {\displaystyle A} is necessarily bounded. A bounded operator A : H → H {\displaystyle A:H\to H} is self-adjoint if ⟨ A x ...

bounded if it is bounded around every point. A family of functions is locally bounded if for any point in their domain all the functions are bounded around...

if it is equal to the algebra of bounded operators which leave invariant each subspace left invariant by every operator in A. This should not be confused...

of 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...

mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space H induced...

are known as sequence spaces. Operators on these spaces are known as sequence transformations. Bounded linear operators over a Banach space form a Banach...

spectrum of bounded operator. The next generalization we consider is that of bounded self-adjoint operators on a Hilbert space. Such operators may have no...

defines a bounded operator from L 2 {\displaystyle L^{2}} to L 2 {\displaystyle L^{2}} . One motivation for the study of Fourier integral operators is the...

product space) is unitarily diagonalizable. Let T {\displaystyle T} be a bounded operator. The following are equivalent. T {\displaystyle T} is normal. T ∗ {\displaystyle...

is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special...

orthonormal basis and A : H → H {\displaystyle A:H\to H} a positive bounded linear operator on H {\displaystyle H} . The trace of A {\displaystyle A} is denoted...

if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map. A special case is also called the bounded inverse...

functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space H {\displaystyle...

In operator theory, a bounded operator T on a Banach space is said to be nilpotent if Tn = 0 for some positive integer n. It is said to be quasinilpotent...

called essentially bounded if h is bounded μ-almost everywhere. An essentially bounded h induces a bounded multiplication operator Th on Lp(μ): ( T h...

Every weakly convergent sequence {xn} is bounded, by the uniform boundedness principle. Conversely, every bounded sequence in a Hilbert space admits weakly...

In operator theory, a bounded operator T: X → Y between normed vector spaces X and Y is said to be a contraction if its operator norm ||T || ≤ 1. This...

to a bounded operator ⁠ M {\displaystyle \mathbf {M} } ⁠ on a separable Hilbert space ⁠ H . {\displaystyle H.} ⁠ Namely, for any bounded operator ⁠ M ...