Function space

In linear algebra

Examples

• In set theory, the set of functions from X to Y may be denoted X → Y or Y^X.
• * As a special case, the power set of a set X may be identified with the set of all functions from X to, denoted 2^X.
• The set of bijections from X to Y is denoted. The factorial notation X! may be used for permutations of a single set X.
• In functional analysis the same is seen for continuous linear transformations, including topologies on the vector spaces in the above, and many of the major examples are function spaces carrying a topology; the best known examples include Hilbert spaces and Banach spaces.
• In functional analysis the set of all functions from the natural numbers to some set X is called a sequence space. It consists of the set of all possible sequences of elements of X.
• In topology, one may attempt to put a topology on the space of continuous functions from a topological space X to another one Y, with utility depending on the nature of the spaces. A commonly used example is the compact-open topology, e.g. loop space. Also available is the product topology on the space of set theoretic functions Y^X. In this context, this topology is also referred to as the topology of pointwise convergence.
• In algebraic topology, the study of homotopy theory is essentially that of discrete invariants of function spaces;
• In the theory of stochastic processes, the basic technical problem is how to construct a probability measure on a function space of paths of the process ;
• In category theory the function space is called an exponential object or map object. It appears in one way as the representation canonical bifunctor; but as functor, of type , it appears as an adjoint functor to a functor of type on objects;
• In functional programming and lambda calculus, function types are used to express the idea of higher-order functions.
• In domain theory, the basic idea is to find constructions from partial orders that can model lambda calculus, by creating a well-behaved cartesian closed category.
• In the representation theory of finite groups, given two finite-dimensional representations and of a group, one can form a representation of over the vector space of linear maps Hom called the Hom representation.

  Functional analysis

• continuous functions endowed with the uniform norm topology
• continuous functions with compact support
• bounded functions
• continuous functions which vanish at infinity
• continuous functions that have continuous first r derivatives.
• smooth functions
• smooth functions with compact support
• real analytic functions
• , for, is the L^p space of measurable functions whose p-norm is finite
• , the Schwartz space of rapidly decreasing smooth functions and its continuous dual, tempered distributions
• compact support in limit topology
• Sobolev space of functions whose weak derivatives up to order k are in
• holomorphic functions
• linear functions
• piecewise linear functions
• continuous functions, compact open topology
• all functions, space of pointwise convergence
• Hardy space
• Hölder space
• Càdlàg functions, also known as the Skorokhod space
• , the space of all Lipschitz functions on that vanish at zero.

  Norm

Footnotes