Functional analysis

Functional analysis studies vector spaces of functions and the operators acting on them, using ideas from linear algebra, topology, and analysis.

Normed vector space: A vector space equipped with a norm that measures the size of each vector. Norms let functional analysis compare functions and define convergence.
Bounded linear operator: A linear map between normed spaces whose outputs are bounded by a constant times the input norm. Boundedness is the normed-space counterpart of continuity for linear maps.
Lp space: A space of measurable functions whose absolute values have finite p-th power integral, with standard endpoint variants. These spaces provide the main settings for convergence and integrability estimates.
Finite-dimensional vector space: A vector space with a finite basis. Finite-dimensional spaces lack many pathologies that arise for function spaces.
Integral equation: An equation in which an unknown function appears inside an integral. Solving integral equations motivated early operator methods and abstract function spaces.
Banach space: A complete normed vector space in which every Cauchy sequence converges. Completeness supports central existence and convergence theorems.
Continuous linear functional: A continuous linear map from a topological vector space to its scalar field. Functionals probe vectors with scalar-valued measurements.
Sobolev space: A function space defined by integrability of a function and some of its weak derivatives. Sobolev spaces make weak solutions of differential equations mathematically precise.
Algebraic dual space: The space of all linear functionals on a vector space, without a continuity requirement. Unlike the continuous dual, it ignores the topology central to functional analysis.
Volterra integral equation: An integral equation in which the integration limit depends on the independent variable. Volterra's equations helped establish systematic study of integral operators.

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