The spaces (\mathcal B(X,Y)) and (\mathcal B(X)) of bounded linear operators

Depends on

• A bounded linear operator between normed spaces
• The space $\mathcal L(V,W)$ of linear maps with pointwise addition and scalar multiplication

Used by

• The Lax--Milgram solution operator has norm at most 1/α Corollary
• Approximable operator Definition
• C k map between Banach spaces Definition
• C star algebra generated by a normal operator Definition
• Complex sector and bounded analytic semigroup Definition
• Fréchet derivative between Banach spaces Definition
• Hilbert–Schmidt operator and Hilbert–Schmidt norm Definition
• Measurable and decomposable operator fields Definition
• Self-adjoint, positive, unitary and normal operators Definition
• Smooth Banach vector bundle and section Definition
• Spectrum and resolvent of a bounded operator Definition
• States, tracial states and faithful normal traces on a von Neumann algebra Definition
• Strong and weak operator topologies Definition
• Strongly continuous semigroup Definition
• The dual space X^* of a normed space and its dual norm Definition
• The Hilbert-space adjoint of a bounded operator Definition
• The negative Sobolev space H⁻¹(Ω) Definition
• Von Neumann algebras and commutants Definition
• Coordinate projections and inclusions on a finite product Banach space Example
• The square of the Volterra operator has zero trace Example
• A coercive Hilbert-complex estimate solves the closed equation Lemma
• A separably acting abelian von Neumann algebra has a self-adjoint generator Lemma
• A sequential approximate identity concentrated near the identity Lemma
• Bounded density and finite-vector transitivity for C*-representations Lemma
• Bounded Hilbert operators form a C star algebra Lemma
• C star state GNS construction, purity and Polish pure-state spaces Lemma
• Composition satisfies ‖ST‖≤‖S‖ ‖T‖ Lemma
• Dominated positive type and positive commutant contractions Lemma
• Measurable Gram-Schmidt and constant-field trivializations on dimension strata Lemma
• Neumann series and small perturbations of bounded inverses Lemma
• Nonscalar commutant contractions and convex decompositions Lemma
• The operator norm is a norm on the space of bounded linear operators Lemma
• Chain sum product and composition rules for Banach derivatives Theorem
• If (Y) is Banach then (B(X,Y)) is Banach Theorem
• Implicit function theorem for Banach spaces Theorem
• Measurable essentially bounded operator fields act decomposably Theorem
• Spectral multiplicity model for separably acting abelian von Neumann algebras Theorem
• The method of continuity for a uniformly estimated affine family of bounded operators Theorem

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Theo Buhler and Dietmar A. Salamon, Functional Analysis (standard reference, not scraped)
• Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis (standard reference, not scraped)