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NORMED VECTOR SPACES In the modern view, functional analysis is seen as the study of Complete Normed Vector Space s over the Real or Complex numbers. Such spaces are called Banach Space s. An important example is a Hilbert Space , where the Norm arises from an Inner Product . These spaces are of fundamental importance in many areas, including the mathematical formulation of Quantum Mechanics . More generally, functional analysis includes the study of Fréchet Space s and other Topological Vector Space s not endowed with a norm.
Hilbert spaces for every Cardinality of the base. Since finite-dimensional Hilbert spaces are fully understood in Linear Algebra , and since Morphisms of Hilbert spaces can always be divided into morphisms of spaces with Aleph-null (ℵ0) dimensionality, functional analysis of Hilbert spaces mostly deals with the unique Hilbert space of dimensionality Aleph-null, and its morphisms. One of the open problems in functional analysis is to prove that every operator on a Hilbert space has a proper Invariant Subspace . Many special cases have already been proven. Banach spaces General Banach Space s are more complicated. There is no clear definition of what would constitute a base, for example. For any real number ''p'' ≥ 1, an example of a Banach space is given by "all Lebesgue-measurable Function s whose Absolute Value 's ''p''-th power has finite integral" (see L''p'' Spaces ). In Banach spaces, a large part of the study involves the linear functionals. The dual of the dual is not always isomorphic to the original space, but there is always a natural Monomorphism from a space into its dual's dual. This is explained in the Dual Space article. Also, the notion of Derivative can be extended to arbitrary functions between Banach spaces. See, for instance, the Fréchet Derivative article. MAJOR AND FOUNDATIONAL RESULTS Important results of functional analysis include:
''See also'': List Of Functional Analysis Topics . FOUNDATIONS OF MATHEMATICS CONSIDERATIONS Most spaces considered in functional analysis have infinite dimension. To show the existence of a Vector Space Basis for such spaces may require Zorn's Lemma . Many very important theorems require the Hahn-Banach Theorem , usually proved using Axiom Of Choice , although the strictly weaker Boolean Prime Ideal Theorem suffices. POINTS OF VIEW Functional analysis in its Present Form includes the following tendencies:
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