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  2. 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.

  3. Weak convergence (Hilbert space) - Wikipedia

    en.wikipedia.org/wiki/Weak_convergence_(Hilbert...

    The Banach–Saks theorem states that every bounded sequence contains a subsequence ... so one obtains the Hilbert space definition of weak convergence. ...

  4. Banach space - Wikipedia

    en.wikipedia.org/wiki/Banach_space

    In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space.Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space.

  5. 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]

  6. 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

  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. Complemented subspace - Wikipedia

    en.wikipedia.org/wiki/Complemented_subspace

    This property characterizes Hilbert spaces within the class of Banach spaces: every infinite dimensional, non-Hilbert Banach space contains a closed uncomplemented subspace, a deep theorem of Joram Lindenstrauss and Lior Tzafriri. [9] [3]

  9. Separable space - Wikipedia

    en.wikipedia.org/wiki/Separable_space

    Every separable metric space is homeomorphic to a subset of the Hilbert cube. This is established in the proof of the Urysohn metrization theorem. Every separable metric space is isometric to a subset of the (non-separable) Banach space l ∞ of all bounded real sequences with the supremum norm; this is known as the