A Course in Functional Analysis

Conway, J. B. (1990). A Course in Functional Analysis. Springer.

Type

Book

Intellectual base

Textbook

Year

1990

DOI

10.1007/978-1-4757-4383-8

ISBN

9781441930927

Link

https://link.springer.com/book/10.1007/978-1-4757-4383-8

Cited from

mathematics

Cited by

15 citations across 15 artifacts.

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

Primes

• Boundedness
  • Boundedness is not the same as compactness, although the two are closely related and famously equivalent in finite-dimensional Euclidean space, as Conway (1990) emphasises in his Course in Functional Analysis when distinguishing the finite-dimensional and infinite-dimensional cases.

    This sourcedevelops the Heine-Borel equivalence (compact iff closed-and-bounded) in finite dimensions and its failure in infinite-dimensional Banach spaces (Riesz's lemma; closed unit ball not compact).

Domain-specific

• Banach Space
  • Conway treats this package as the basic setting for functional analysis and bounded linear operators.
• C space
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Compact Operator
  • A compact operator is a linear operator \(T:X\to Y\) that maps every bounded subset of \(X\) to a relatively compact subset of \(Y\); equivalently, the closure of \(T(B_X)\), the image of the closed unit ball, is compact.
• Crinkled Arc
  • is immediate because a chord is the indicator of an interval and indicators of disjoint intervals are orthogonal.
• Fundamental theorem of Hilbert spaces
  • The theorem states that a Hausdorff pre-Hilbert space is complete exactly when every continuous functional is represented by inner product with a vector, equivalently its canonical map onto the anti-dual is surjective.
• Holomorphic functional calculus
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Hyponormal operator
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Local boundedness
  • … between constitutive identity and a convenient observable transfers from The function one over x is locally bounded on the open domain excluding zero but is not globally bounded there. to A proof distinguishes a locally bounded family with one bound per neighborhood from each member being locally bounded separately..
• Operator topologies
  • Operator topologies formalize different strengths of saying a sequence or net of operators converges.
• Partial isometry
  • … identity and a convenient observable transfers from The unilateral shift is an isometry and hence a partial isometry with the whole domain as initial subspace. to An operator proof identifies closed initial and final spaces and distinguishes partial isometry from a norm-preserving map on only selected vectors..
• Space of continuous functions on a compact space
  • The space \(C(X,\mathbb F)\) consists of all continuous maps from compact Hausdorff \(X\) to \(\mathbb F\), with \(\lVert f\rVert_\infty=\sup_{x\in X}|f(x)|\) and pointwise vector-space and algebra operations.
• Total subset
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Uniform norm
  • … canonical instance directly demonstrates that the function is bounded in the declared value norm and the norm equals the least upper bound of its pointwise magnitudes over the stated domain. to An applied instance preserves the same invariant under a changed scale, notation, jurisdiction, dataset, or implementation..
• Unitary operator
  • Infinite-dimensional boundedness and surjectivity matter; an isometry need not be unitary if its range is proper.

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