In topology, the **Tietze extension theorem** (also known as the Tietze–Urysohn–Brouwer extension theorem) states that continuous functions on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness if necessary.

If is a normal space and

is a continuous map from a closed subset of into the real numbers carrying the standard topology, then there exists a * continuous extension * of to that is, there exists a map

continuous on all of with for all Moreover, may be chosen such that

that is, if is bounded then may be chosen to be bounded (with the same bound as ).

L. E. J. Brouwer and Henri Lebesgue proved a special case of the theorem, when is a finite-dimensional real vector space. Heinrich Tietze extended it to all metric spaces, and Pavel Urysohn proved the theorem as stated here, for normal topological spaces.^{ [1] }^{ [2] }

This theorem is equivalent to Urysohn's lemma (which is also equivalent to the normality of the space) and is widely applicable, since all metric spaces and all compact Hausdorff spaces are normal. It can be generalized by replacing with for some indexing set any retract of or any normal absolute retract whatsoever.

If is a metric space, a non-empty subset of and is a Lipschitz continuous function with Lipschitz constant then can be extended to a Lipschitz continuous function with same constant This theorem is also valid for Hölder continuous functions, that is, if is Hölder continuous function with constant less than or equal to then can be extended to a Hölder continuous function with the same constant.^{ [3] }

Another variant (in fact, generalization) of Tietze's theorem is due to H.Tong and Z. Ercan:^{ [4] } Let be a closed subset of a normal topological space If is an upper semicontinuous function, a lower semicontinuous function, and a continuous function such that for each and for each , then there is a continuous extension of such that for each This theorem is also valid with some additional hypothesis if is replaced by a general locally solid Riesz space.^{ [4] }

- Blumberg theorem – Any real function on R admits a continuous restriction on a dense subset of R
- Hahn–Banach theorem – Theorem on extension of bounded linear functionals
- Whitney extension theorem – Partial converse of Taylor's theorem

In mathematics, more specifically in functional analysis, a **Banach space** is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.

In mathematics, specifically general topology, **compactness** is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "holes" or "missing endpoints", i.e. that the space not exclude any "limiting values" of points. For example, the "unclosed" interval (0,1) would not be compact because it excludes the "limiting values" of 0 and 1, whereas the closed interval [0,1] *would* be compact. Similarly, the space of rational numbers is not compact because it has infinitely many "holes" corresponding to the irrational numbers, and the space of real numbers is not compact either because it excludes the limiting values and . However, the *extended* real number line *would* be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in Euclidean space, but may be inequivalent in other topological spaces.

In mathematics, a **continuous function** is a function such that a **continuous variation** of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as *discontinuities*. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A **discontinuous function** is a function that is *not continuous*. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity.

The **Hahn–Banach theorem** is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space "interesting". Another version of the Hahn–Banach theorem is known as the **Hahn–Banach separation theorem** or the hyperplane separation theorem, and has numerous uses in convex geometry.

In mathematics, a **metric space** is a set together with a metric on the set. The metric is a function that defines a concept of *distance* between any two members of the set, which are usually called points. The metric satisfies the following properties:

In mathematics, a topological space is called **separable** if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

In topology and related branches of mathematics, a **normal space** is a topological space *X* that satisfies **Axiom T _{4}**: every two disjoint closed sets of

In topology, **Urysohn's lemma** is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a continuous function.

In mathematical analysis, **Lipschitz continuity**, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the *Lipschitz constant* of the function. For instance, every function that has bounded first derivatives is Lipschitz continuous.

**Distributions**, also known as **Schwartz distributions** or **generalized functions**, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative.

In mathematical analysis, **semicontinuity** is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function is **upper****semicontinuous** at a point if, roughly speaking, the function values for arguments near are not much higher than

In functional analysis and related areas of mathematics, **Fréchet spaces**, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces. All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically *not* Banach spaces.

In mathematical analysis, a family of functions is **equicontinuous** if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus *sequences* of functions.

In functional analysis and related areas of mathematics, **locally convex topological vector spaces** (**LCTVS**) or **locally convex spaces** are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

In functional analysis and related branches of mathematics, the **Banach–Alaoglu theorem** states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proof identifies the unit ball with the weak-* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact.

The **Arzelà–Ascoli theorem** is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded interval has a uniformly convergent subsequence. The main condition is the equicontinuity of the family of functions. The theorem is the basis of many proofs in mathematics, including that of the Peano existence theorem in the theory of ordinary differential equations, Montel's theorem in complex analysis, and the Peter–Weyl theorem in harmonic analysis and various results concerning compactness of integral operators.

The **Katětov–Tong insertion theorem** is a theorem of point-set topology proved independently by Miroslav Katětov and Hing Tong in the 1950s. The theorem states the following:

The **maximum theorem** provides conditions for the continuity of an optimized function and the set of its maximizers with respect to its parameters. The statement was first proven by Claude Berge in 1959. The theorem is primarily used in mathematical economics and optimal control.

In mathematics, the **Blumberg theorem** states that for any real function there is a dense subset of such that the restriction of to is continuous.

In mathematical analysis, the **spaces of test functions and distributions** are topological vector spaces (TVSs) that are used in the definition and application of distributions. Test functions are usually infinitely differentiable complex-valued functions on a non-empty open subset that have compact support. The space of all test functions, denoted by is endowed with a certain topology, called the *canonical LF-topology*, that makes into a complete Hausdorff locally convex TVS. The strong dual space of is called * the space of distributions on * and is denoted by where the "" subscript indicates that the continuous dual space of denoted by is endowed with the strong dual topology.

- ↑ "Urysohn-Brouwer lemma",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - ↑ Urysohn, Paul (1925), "Über die Mächtigkeit der zusammenhängenden Mengen",
*Mathematische Annalen*,**94**(1): 262–295, doi:10.1007/BF01208659, hdl: 10338.dmlcz/101038 . - ↑ McShane, E. J. (1 December 1934). "Extension of range of functions".
*Bulletin of the American Mathematical Society*.**40**(12): 837–843. doi: 10.1090/S0002-9904-1934-05978-0 . - 1 2 Zafer, Ercan (1997). "Extension and Separation of Vector Valued Functions" (PDF).
*Turkish Journal of Mathematics*.**21**(4): 423–430.

- Munkres, James R. (2000).
*Topology*(Second ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260.

- Weisstein, Eric W. "Tietze's Extension Theorem." From MathWorld
- Mizar system proof: http://mizar.org/version/current/html/tietze.html#T23
- Bonan, Edmond (1971), "Relèvements-Prolongements à valeurs dans les espaces de Fréchet",
*Comptes Rendus de l'Académie des Sciences, Série I*,**272**: 714–717.

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.