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

standard basis, and one defines the corresponding induced norm or operator norm or subordinate norm on the space K m × n {\displaystyle K^{m\times n}} of...

norm). The set of Hilbert–Schmidt operators is closed in the norm topology if, and only if, H is finite-dimensional. Every Hilbert–Schmidt operator T :...

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

every x in U. Bounded operators form a vector space. On this vector space we can introduce a norm that is compatible with the norms of U and V: ‖ A ⁡ ‖...

Matrix norm, a map that assigns a length or size to a matrix Operator norm, a map that assigns a length or size to any operator in a function space Norm (abelian...

In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance...

Schatten norm (or Schatten–von-Neumann norm) arises as a generalization of p-integrability similar to the trace class norm and the Hilbert–Schmidt norm. Let...

reference to algebras of operators on a separable Hilbert space, endowed with the operator norm topology. In the case of operators on a Hilbert space, the...

Theorems 1 and 2 below.) The dual norm is a special case of the operator norm defined for each (bounded) linear map between normed vector spaces. Since the ground...

operator 2-norm. One can easily verify the relationship between the Ky Fan 1-norm and singular values. It is true in general, for a bounded operator ⁠...

one can say that the weak-operator and σ-weak topologies agree on norm-bounded sets in B(H): Every trace-class operator is of the form S = ∑ i λ i u...

inequality for integral operators, is a bound on the L p → L q {\displaystyle L^{p}\to L^{q}} operator norm of an integral operator in terms of L r {\displaystyle...

mathematics, a compact operator is a linear operator T : X → Y {\displaystyle T:X\to Y} , where X , Y {\displaystyle X,Y} are normed vector spaces, with...

operator norm? Every finite-dimensional reflexive algebra is hyper-reflexive. However, there are examples of infinite-dimensional reflexive operator algebras...

logarithmic norm is a real-valued functional on operators, and is derived from either an inner product, a vector norm, or its induced operator norm. The logarithmic...

linear operators (and thus bounded operators) whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm. The...

is a bounded linear operator on the normed vector space X {\displaystyle X} . If the Neumann series converges in the operator norm, then I − T {\displaystyle...

some operator T on X. This could have several different meanings: If ‖ T n − T ‖ → 0 {\displaystyle \|T_{n}-T\|\to 0} , that is, the operator norm of T...

formula, also holds for bounded linear operators: letting ‖ ⋅ ‖ {\displaystyle \|\cdot \|} denote the operator norm, we have ρ ( A ) = lim k → ∞ ‖ A k ‖...

. In this case, its operator norm is equal to ‖ f ‖ ∞ {\displaystyle \|f\|_{\infty }} . The adjoint of a multiplication operator T f {\displaystyle T_{f}}...

set of Fredholm operators from X to Y is open in the Banach space L(X, Y) of bounded linear operators, equipped with the operator norm, and the index is...

spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue...

linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces...

(σ1(T), σ2(T), …). The largest singular value σ1(T) is equal to the operator norm of T (see Min-max theorem). If T acts on Euclidean space R n {\displaystyle...

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...

closure of finite-rank operators (representable by finite-dimensional matrices) in the topology induced by the operator norm. As such, results from matrix...

transpose, of an operator A : E → F {\displaystyle A:E\to F} , where E , F {\displaystyle E,F} are Banach spaces with corresponding norms ‖ ⋅ ‖ E , ‖ ⋅ ‖...

Conversely, if an operator is bounded, then it is continuous. The space of such bounded linear operators has a norm, the operator norm given by ‖ A ‖ =...

composition as multiplication and the operator norm as norm) is a unital Banach algebra. The set of all compact operators on E {\displaystyle E} is a Banach...