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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 Spaces . 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. The notion of Derivative is extended to arbitrary functions between Banach spaces. It turns out that the derivative of a function at a certain point is really a continuous linear map. MAJOR AND FOUNDATIONAL RESULTS These are important results of functional analysis:
''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 , which relies on the Axiom Of Choice that is strictly weaker than the Boolean Prime Ideal Theorem . POINTS OF VIEW Functional analysis as it Currently stands includes a number of directions:
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