Function space

In

This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (November 2017) (Learn how and when to remove this message)

Let F be a field and let X be any set. The functions X → F can be given the structure of a vector space over F where the operations are defined pointwise, that is, for any f, g : X → F, any x in X, and any c in F, define $${\displaystyle {\begin{aligned}(f+g)(x)&=f(x)+g(x)\\(c\cdot f)(x)&=c\cdot f(x)\end{aligned}}}$$ When the domain X has additional structure, one might consider instead the

The cardinal dimension of a function space with no extra structure can be found by the Erdős–Kaplansky theorem.

Examples

Function spaces appear in various areas of mathematics:

• 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 {0, 1}, denoted 2^X.
• The set of bijections from X to Y is denoted ${\displaystyle X\leftrightarrow Y}$. 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
  sequences
  of elements of X.
• In
  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 (functions of time);
• In
  adjoint functor
  to a functor of type (-×X) on objects;
• In functional programming and lambda calculus, function types are used to express the idea of higher-order functions.
• In
  partial orders that can model lambda calculus, by creating a well-behaved Cartesian closed category
  .
• In the
  Hom representation.^[1]

Functional analysis

normed spaces

of finite dimension. Here we use the real line as an example domain, but the spaces below exist on suitable open subsets ${\displaystyle \Omega \subseteq \mathbb {R} ^{n}}$

• ${\displaystyle C(\mathbb {R} )}$
  continuous functions
  endowed with the uniform norm topology
• ${\displaystyle C_{c}(\mathbb {R} )}$ continuous functions with compact support
• ${\displaystyle B(\mathbb {R} )}$ bounded functions
• ${\displaystyle C_{0}(\mathbb {R} )}$ continuous functions which vanish at infinity
• ${\displaystyle C^{r}(\mathbb {R} )}$ continuous functions that have continuous first r derivatives.
• ${\displaystyle C^{\infty }(\mathbb {R} )}$
  smooth functions
• ${\displaystyle C_{c}^{\infty }(\mathbb {R} )}$
  smooth functions with compact support
• ${\displaystyle C^{\omega }(\mathbb {R} )}$ real analytic functions
• ${\displaystyle L^{p}(\mathbb {R} )}$, for ${\displaystyle 1\leq p\leq \infty }$, is the L^p space of measurable functions whose p-norm ${\textstyle \|f\|_{p}=\left(\int _{\mathbb {R} }|f|^{p}\right)^{1/p}}$ is finite
• ${\displaystyle {\mathcal {S}}(\mathbb {R} )}$, the
  smooth functions
  and its continuous dual, ${\displaystyle {\mathcal {S}}'(\mathbb {R} )}$
  tempered distributions
• ${\displaystyle D(\mathbb {R} )}$ compact support in limit topology
• ${\displaystyle W^{k,p}}$ Sobolev space of functions whose weak derivatives up to order k are in ${\displaystyle L^{p}}$
• ${\displaystyle {\mathcal {O}}_{U}}$ 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
• ${\displaystyle {\text{Lip}}_{0}(\mathbb {R} )}$, the space of all
  Lipschitz
  functions on ${\displaystyle \mathbb {R} }$ that vanish at zero.

Norm

If y is an element of the function space ${\displaystyle {\mathcal {C}}(a,b)}$ of all

${\displaystyle \|y\|_{\infty }}$

defined on ${\displaystyle {\mathcal {C}}(a,b)}$ is the maximum

$${\displaystyle \|y\|_{\infty }\equiv \max _{a\leq x\leq b}|y(x)|\qquad {\text{where}}\ \ y\in {\mathcal {C}}(a,b)}$$

is called the uniform norm or supremum norm ('sup norm').

Bibliography

• Kolmogorov, A. N., & Fomin, S. V. (1967). Elements of the theory of functions and functional analysis. Courier Dover Publications.
• Stein, Elias; Shakarchi, R. (2011). Functional Analysis: An Introduction to Further Topics in Analysis. Princeton University Press.

See also

• List of mathematical functions
• Clifford algebra
• Tensor field
• Spectral theory
• Functional determinant

References