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  2. Type and cotype of a Banach space - Wikipedia

    en.wikipedia.org/wiki/Type_and_cotype_of_a...

    In functional analysis, the type and cotype of a Banach space are a classification of Banach spaces through probability theory and a measure, how far a Banach space from a Hilbert space is. The starting point is the Pythagorean identity for orthogonal vectors ( e k ) k = 1 n {\displaystyle (e_{k})_{k=1}^{n}} in Hilbert spaces

  3. Hilbert space - Wikipedia

    en.wikipedia.org/wiki/Hilbert_space

    Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space. A Hilbert space is a special case of a Banach space. Hilbert spaces were studied beginning in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz.

  4. Spectral theory - Wikipedia

    en.wikipedia.org/wiki/Spectral_theory

    This definition applies to a Banach space, but of course other types of space exist as well; for example, topological vector spaces include Banach spaces, but can be more general. [12] [13] On the other hand, Banach spaces include Hilbert spaces, and it is these spaces that find the greatest application and the richest theoretical results. [14]

  5. Delta-convergence - Wikipedia

    en.wikipedia.org/wiki/Delta-convergence

    In mathematics, Delta-convergence, or Δ-convergence, is a mode of convergence in metric spaces, weaker than the usual metric convergence, and similar to (but distinct from) the weak convergence in Banach spaces. In Hilbert space, Delta-convergence and weak convergence coincide. For a general class of spaces, similarly to weak convergence ...

  6. Approximation property - Wikipedia

    en.wikipedia.org/wiki/Approximation_property

    The construction of a Banach space without the approximation property earned Per Enflo a live goose in 1972, which had been promised by Stanisław Mazur (left) in 1936. [ 1 ] In mathematics , specifically functional analysis , a Banach space is said to have the approximation property (AP) , if every compact operator is a limit of finite-rank ...

  7. Weak convergence - Wikipedia

    en.wikipedia.org/wiki/Weak_convergence

    Weak convergence (Hilbert space) of a sequence in a Hilbert space more generally, convergence in weak topology in a Banach space or a topological vector space Topics referred to by the same term

  8. Decomposition of spectrum (functional analysis) - Wikipedia

    en.wikipedia.org/wiki/Decomposition_of_spectrum...

    For a Banach space, T* denotes the transpose and σ(T*) = σ(T). For a Hilbert space, T* normally denotes the adjoint of an operator T ∈ B(H), not the transpose, and σ(T*) is not σ(T) but rather its image under complex conjugation. For a self-adjoint T ∈ B(H), the Borel functional calculus gives additional ways to break up the spectrum ...

  9. Operator topologies - Wikipedia

    en.wikipedia.org/wiki/Operator_topologies

    The weak (Banach space) topology is σ(B(H), B(H) *), in other words the weakest topology such that all elements of the dual B(H) * are continuous. It is the weak topology on the Banach space B(H). It is stronger than the ultraweak and weak operator topologies.