Banach space

Depends on

• A norm on a real vector space, the induced metric, and the dictionary with the metric axioms
• Complete metric space: every Cauchy sequence converges in the space

Used by

• A one-sided comparison of two complete norms makes them equivalent Corollary
• Compact operator iff approximation numbers tend to zero Corollary
• Every finite-dimensional normed space is Banach Corollary
• Lambda identity minus compact has index zero Corollary
• Spectrum of a compact operator is countable with only zero as possible accumulation Corollary
• A closed subspace of ell-infinity that is not complemented Counterexample
• A complete domain is necessary for sequential uniform boundedness Counterexample
• Nearest-point maps to convex sets need not be linear Counterexample
• Weakly measurable need not be strongly measurable Counterexample
• Approximable operator Definition
• Approximation property and bounded approximation property Definition
• Banach algebra valued contour integral Definition
• Banach star-algebra without a required unit Definition
• Banach-valued simple function and integral Definition
• Banach-valued vector measure and variation Definition
• C k map between Banach spaces Definition
• C star algebra Definition
• Completion of a normed space Definition
• Complex sector and bounded analytic semigroup Definition
• Countable base Banach manifold and smooth map Definition
• Dissipative operator Definition
• Fréchet derivative between Banach spaces Definition
• Fredholm operator cokernel and index Definition
• Hilbert space Definition
• Infinitesimal generator of a C0-semigroup Definition
• Proper, coercive and weakly lower semicontinuous extended-real functionals Definition
• Reflexivity is surjectivity of the canonical map Definition
• Resolvent and spectrum of a closed operator on a Banach space Definition
• Schauder basis and coordinate functionals Definition
• Sectorial operator with the semigroup sign convention Definition
• Smooth Banach vector bundle and section Definition
• Spectrum and resolvent of a bounded operator Definition
• Strongly continuous semigroup Definition
• Uniformly convex Banach space Definition
• Unital Banach algebra Definition
• A projection with finite-dimensional kernel is Fredholm Example
• A regular level set in a Banach space Example
• C_b(X) is Banach for the supremum norm Example
• Coordinate partial sums on c₀ Example
• Diagonal operator on ell p is compact iff diagonal tends to zero Example

…and 79 more results.

Dependency tree · two levels

19 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)
• Gerald Teschl, Topics in Real and Functional Analysis (standard reference, not scraped)