Functional Analysis

Rudin, W. (1991). Functional Analysis. McGraw-Hill.

Type

Book

Intellectual base

Textbook

Year

1991

ISBN

9780070542365

Link

https://openlibrary.org/books/OL1853956Mmatched

Cited from

computer_science

Cited by

12 citations across 12 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Higher Order Function
  • In mathematics it is operators on functions: differentiation maps a function to its derivative, the Fourier and Laplace transforms map functions to functions, and functionals map functions to scalars — the whole field of functional analysis studies higher-order operations on function spaces.

    This sourceStandard text treating operators and functionals as higher-order maps on function spaces — differentiation, integral transforms, and linear functionals.

Domain-specific

• Antilinear Map
  • It is a precise semilinear rule: over the underlying real vector spaces, (A) is linear, while multiplication by (i) reverses sign, (A(iv)=-iA(v)).
• C space
  • … distinction between constitutive identity and a convenient observable transfers from The sequence (1+1/n) belongs to c and converges to 1 but does not belong to c0. to Arguments distinguish c from the space of continuous functions often denoted C and track the nonzero limit component when comparing its dual with l1..
• Compactness
  • the failure modes — what destroys the property, diagnosing the pathological branch: non-closedness, unboundedness, and infinite-dimensionality (the norm-closed unit ball of an infinite-dimensional Banach space is not compact

    Supported in partVerified against a saved copy of the source

    “called the weak*-topology of xX*”

    Read by an automated reader; not a human review. How support was checked

• Fréchet space
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Fundamental theorem of Hilbert spaces
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Holomorphic functional calculus
  • … holomorphic neighborhood, oriented admissible contour, resolvent and normalization, integral definition, independence proof, algebra-homomorphism law and spectral mapping claim are explicit. to An applied instance preserves the invariant under changed notation, scale, dataset, jurisdiction, or implementation..
• Lomonosov's invariant subspace theorem
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Space of continuous functions on a compact space
  • … scalar function on the interval and is complete in \(\lVert\cdot\rVert_\infty\). to For compact Hausdorff \(X\), every continuous linear functional on \(C(X)\) is represented by integration against a finite regular signed or complex Borel measure under the Riesz–Markov–Kakutani theorem. demonstrates that continuity.
• Topological homomorphism
  • … target topological vector spaces, continuous linear map, kernel and quotient topology, image and subspace topology and induced topological isomorphism or open-onto-image condition are explicit. to An applied instance preserves the same invariant under changed scale, notation, jurisdiction, dataset, or implementation..
• Topological Space
  • Functional analysis — topological vector spaces, locally convex spaces, and Banach/Hilbert spaces are topological spaces with extra structure, with the weak and weak-* topologies central to duality

    This sourceTopological vector spaces, locally convex and Banach spaces, and the weak and weak* topologies that organize duality theory.

    SupportedVerified against the source

    Read by an automated reader; not a human review. How support was checked

• Total subset
  • A total subset provides enough vectors to approximate every point of the ambient topological vector space by finite linear combinations.

Verification

Does it exist? Confirmed. This work's ISBN resolves to a registered record, which fixes its identity. That is all it fixes.

Does it back the claim? Read against the text for 2 of 12 citations: 1 supported, 1 supported in part. Each verdict is shown under its citation below, with what in the work backs the sentence.

Support is checked per citation rather than per work — the same source can be cited soundly in one article and wrongly in another. Per-citation recording began recently, so a citation with no recorded check is a gap in the record rather than evidence it went unchecked.

See how references were verified.

Registry ID ref:6efbe4dec347 · see in the full table