C ₀ function

This article deals with continuous functions that vanish at infinity. For differentiable functions with a compact carrier, see test function .

Normal distributions are described by continuous density functions that never take on the value 0, but disappear at infinity.

The density function of a two-dimensional normal distribution also disappears at infinity: any fixed height is only exceeded within a circle with a finite radius.

In mathematics , a function is a continuous function that, when viewed graphically, vanishes into infinity. The set of all functions forms a standardized space . ${\displaystyle C_{0}}$ ${\displaystyle C_{0}}$

definition

Let be a topological space (for example the real numbers or the ). A function with or vanishes in infinity if and only if for every neighborhood of zero a compact subset (in the case of a bounded subset ) of exists, so that the image is a subset of . If it is also continuous, it is called a function . The set of all these functions is denoted by or - if no misunderstandings are to be feared - with . ${\displaystyle X}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle f\colon X\to \mathbb {K} }$ ${\displaystyle \mathbb {K} =\mathbb {R} }$ ${\displaystyle \mathbb {K} =\mathbb {C} }$ ${\displaystyle U\subset \mathbb {K} }$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle K}$ ${\displaystyle X}$ ${\displaystyle f(X\setminus K)}$ ${\displaystyle U}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle C_{0}}$ ${\displaystyle C_{0}(X,\mathbb {K} )}$ ${\displaystyle C_{0}(X)}$

One considers this term only for locally compact Hausdorff spaces , because for non-Hausdorff spaces the concept of compactness is rather pathological and for a not locally compact Hausdorff space all points without a compact environment would have to be the zeros of each function: no environment of the respective point is in one Contains compact language, so in every environment the function assumes values ​​arbitrarily close to zero, due to the continuity the point is thus zero. Thus in every not locally compact, homogeneous Hausdorff space - a typical example would be an infinitely dimensional normed space ( see also Riesz's compactness theorem ) - every function would be 0. Therefore, in the following, always be locally compact and Hausdorffsch. ${\displaystyle C_{0}}$ ${\displaystyle C_{0}}$ ${\displaystyle X}$

More abstract definition: ( let the space of all continuous functions on ) is a -function if and only if is compact or the image filter under the filter that is generated by the complements of compact subsets of converges to 0. ${\displaystyle f\in C(X)}$ ${\displaystyle C(X)}$ ${\displaystyle X}$ ${\displaystyle C_{0}}$ ${\displaystyle X}$ ${\displaystyle f}$ ${\displaystyle {\mathfrak {F}}}$ ${\displaystyle X}$

Examples

• Every real function that is outside a bounded set 0 vanishes at infinity. If it is continuous, it is a function. ${\displaystyle C_{0}}$
• More generally, every continuous function with a compact carrier is a function. ${\displaystyle C_{0}}$
• In particular, every continuous function in a compact space is also a function. ${\displaystyle C_{0}}$
• The density function of a normal distribution , essentially plus scales and shifts, is a function. ${\displaystyle x\mapsto \exp(-x^{2})}$ ${\displaystyle C_{0}}$
• A function is a -function if and only if it is continuous and . ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$ ${\displaystyle C_{0}}$ ${\displaystyle \textstyle \lim _{x\to +\infty }f(x)=\lim _{x\to -\infty }f(x)=0}$
• Be discreet . The compact sets are then exactly the finite sets. That is, a function is if and only if it only takes a finite number of times for each value that is greater than in terms of absolute value . The requirement for continuity is not a restriction in this case, since every function is continuous on a discrete space. One then denotes as , where is an arbitrary set, i.e. the topological structure is not assumed. For this is precisely the sequence space of all zero sequences . ${\displaystyle X}$ ${\displaystyle C_{0}(X)}$ ${\displaystyle \varepsilon }$ ${\displaystyle \varepsilon }$ ${\displaystyle C_{0}(X)}$ ${\displaystyle c_{0}(X)}$ ${\displaystyle X}$ ${\displaystyle X=\mathbb {N} }$ ${\displaystyle c_{0}}$
• If not compact, the constant function with value 1 is continuous, but not . ${\displaystyle X}$ ${\displaystyle C_{0}}$

Standardized space

The sum of two functions is a function, as is the point-wise product with a real or complex number. This forms a vector space. In addition, every function is restricted: Let it be a compact form whose existence is guaranteed by definition, so that outside of this compact form the function becomes smaller in absolute terms than any positive real number. Thus, the function outside of the compact is limited. It is also restricted within the compact system, since a continuous function is always restricted to a compact system. Thus, the function is limited to the whole room. Therefore, the room can be equipped with the supreme norm . thus becomes a standardized space . This is complete with regard to the norm and thus a Banach space . It can be used as a closed sub-vector space , i.e. H. as a sub-annex space, the space of all limited functions with the supremum norm. ${\displaystyle C_{0}}$ ${\displaystyle C_{0}}$ ${\displaystyle C_{0}(X)}$ ${\displaystyle C_{0}}$ ${\displaystyle K}$ ${\displaystyle C_{0}(X)}$ ${\displaystyle \|\cdot \|_{\infty }}$ ${\displaystyle C_{0}(X)}$

The continuous functions with compact support form a dense subspace of . It is easy to see that every uniform limit (i.e. with respect to the supremum norm) of functions is a function. Conversely, one exploits the fact that in a locally compact space there is a continuous function with compact support for every compact subset , which assumes the value 1 ( see also decomposition of one ): Let , and for . Choose a function as described above. Then is and . Example: In the case of sequence spaces , given the discrete topology, this is the set of all sequences that ultimately only have the value 0. This is close to the space of the zero sequences . ${\displaystyle C_{c}(X)}$ ${\displaystyle C_{0}(X)}$ ${\displaystyle C_{c}}$ ${\displaystyle C_{0}}$ ${\displaystyle K}$ ${\displaystyle g\colon X\to [0,1]}$ ${\displaystyle K}$ ${\displaystyle f\in C_{0}(X)}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle |f(x)|<\varepsilon }$ ${\displaystyle x\notin K}$ ${\displaystyle g}$ ${\displaystyle fg\in C_{c}(X)}$ ${\displaystyle \|fg-f\|_{\infty }<\varepsilon }$ ${\displaystyle C_{c}(\mathbb {N} )}$ ${\displaystyle \mathbb {N} }$ ${\displaystyle c_{00}}$ ${\displaystyle c_{0}}$

algebra

The point-wise product of two -functions is in turn a -function, thus forming a commutative algebra . Even the product of a -function with any bounded continuous function is in turn a -function, thus forming a closed ideal in the space of bounded functions. Obviously holds for the inequality . Hence it is a Banach algebra . Let now be the point-wise complex conjugation of or just in the real case, then applies . Therefore even a commutative C * -algebra forms with this involution . According to Gelfand-Neumark's theorem , every commutative, complex C * -algebra is isomorphic to a space for a locally compact Hausdorff space . It's an ideal in . ${\displaystyle C_{0}}$ ${\displaystyle C_{0}}$ ${\displaystyle C_{0}(X)}$ ${\displaystyle C_{0}}$ ${\displaystyle C_{0}}$ ${\displaystyle f,g\in C_{0}(X)}$ ${\displaystyle \|f\cdot g\|_{\infty }\leq \|f\|_{\infty }\|g\|_{\infty }}$ ${\displaystyle C_{0}(X)}$ ${\displaystyle f^{*}}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle \|f^{*}f\|_{\infty }=\|f\|_{\infty }^{2}}$ ${\displaystyle C_{0}(X)}$ ${\displaystyle C_{0}(X,\mathbb {C} )}$ ${\displaystyle X}$ ${\displaystyle C_{c}(X)}$ ${\displaystyle C_{0}(X)}$

Alexandroff compactification

→ Main article : Alexandroff compactification

Do n't be compact in this section . The filter defined above does not converge on , but its image filter does . The Alexandroff compactification of space is now space supplemented by an infinitely distant point towards which this filter may converge. Its environments are precisely the elements of combined with , . This space is compact and each function can be continued with a function . This continuation is continuous, because the image of converges to 0, the image of . Conversely, it can be shown that a function on a locally compact Hausdorff space can be continued to a continuous function on the Alexandroff compactification precisely if it has the form with and . ${\displaystyle X}$ ${\displaystyle {\mathfrak {F}}}$ ${\displaystyle X}$ ${\displaystyle {\bar {X}}}$ ${\displaystyle X}$ ${\displaystyle \infty }$ ${\displaystyle {\mathfrak {F}}}$ ${\displaystyle \{\infty \}}$ ${\displaystyle {\bar {\mathfrak {F}}}:=\{U\cup \{\infty \}\mid U\in {\mathfrak {F}}\}}$ ${\displaystyle f\in C_{0}(X)}$ ${\displaystyle {\bar {f}}\colon X\to \mathbb {K} }$ ${\displaystyle f(\infty )=0}$ ${\displaystyle {\bar {\mathfrak {F}}}}$ ${\displaystyle \infty }$ ${\displaystyle k+f}$ ${\displaystyle k\in \mathbb {K} }$ ${\displaystyle f\in C_{0}(X)}$

Stone-Weierstrass theorem and Separability

By continuing to the Alexandroff compactification, the Stone-Weierstrass theorem can be transferred from continuous functions in a compact space to the functions in a locally compact space: It applies that every closed, point-separating , involutive (i.e. also closed under the conjugation) Is subalgebra of either itself or a subalgebra for one . So it follows again that it is close in . ${\displaystyle C_{0}}$ ${\displaystyle C_{0}(X)}$ ${\displaystyle C_{0}(X)}$ ${\displaystyle \{f\in C_{0}(X)\mid f(x_{0})=0\}}$ ${\displaystyle x_{0}\in X}$ ${\displaystyle C_{c}(X)}$ ${\displaystyle C_{0}(X)}$

Using Stone-Weierstrass theorem it can be shown that it is separable if and only if the space satisfies the second axiom of countability . First, fulfill the second axiom of countability. One now selects a countable basis of the topology from relatively compact subsets; this is possible by simply deleting all not relatively compact elements from a countable basis. For with choose a continuous function that takes the value 1 on and 0 outside of . In this way one obtains a countable set of -functions, which can be supplemented to a countable - (or - in the complex case) sub-algebra. This is point-separating and at no point always 0, because there are surroundings of with that do not contain. The corresponding function then assumes the value 1 and the value 0. In order for this subalgebra is after the Stone-Weierstrass theorem dense in . Conversely, it follows from the separability of that the second axiom of countability is satisfied: Let a countable dense subset of be given. This separates points from closed sets, because for each point and each closed set there is a function that assumes the value 0 and the value 1, i.e. also a function in that is smaller than in terms of amount and greater than in terms of amount . thus carries the initial topology with respect to . A countable sub-base and thus a countable base result from the archetypes with regard to the functions in a countable base in . ${\displaystyle C_{0}(X)}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {B}}}$ ${\displaystyle {\overline {U}}\subset V}$ ${\displaystyle U,V\in {\mathcal {B}}}$ ${\displaystyle {\overline {U}}}$ ${\displaystyle V}$ ${\displaystyle C_{c}}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Q} [i]}$ ${\displaystyle x,y\in X}$ ${\displaystyle U,V\in {\mathcal {B}}}$ ${\displaystyle x}$ ${\displaystyle {\overline {U}}\subset V}$ ${\displaystyle y}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle C_{0}(X)}$ ${\displaystyle C_{0}(X)}$ ${\displaystyle X}$ ${\displaystyle S}$ ${\displaystyle C_{0}(X)}$ ${\displaystyle x}$ ${\displaystyle A\not \ni x}$ ${\displaystyle C_{0}}$ ${\displaystyle A}$ ${\displaystyle x}$ ${\displaystyle S}$ ${\displaystyle A}$ ${\displaystyle \textstyle {\frac {1}{2}}}$ ${\displaystyle x}$ ${\displaystyle \textstyle {\frac {1}{2}}}$ ${\displaystyle X}$ ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle \mathbb {K} }$

Dual space

According to a variant of Riesz-Markow's theorem of representation , the positive (linear) functionals in the space of continuous functions with compact carriers correspond exactly to the space of regular measures , i.e. H. for every positive functional there is a regular measure , so that the functional is nothing other than the mapping . Such a functional can be continued to a continuous functional if and only if it is continuous, that is, limited, because the completion of and the continuability result directly from the universal property of completion. This continuation is clear. Conversely, each positive continuous functional can of course be restricted to one to . The positive continuous functionals on correspond exactly to the regular, finite measures on . Each element of the dual space , i.e. H. every continuous functional can be represented as the difference between two positive continuous functionals (in the complex case positive and negative complex components are added). These correspond to finite regular measures, which can in turn be put together to a signed measure (or in the complex case a complex measure ) using the Hahn-Jordan decomposition . The dual space thus corresponds exactly to the regular, finite signed or complex measures. More precisely fitting MAN these measurements with the variation norm of (for positive mass is the just the measure of the entire space), they form a Banach space, the isomorphic to is finite signed or complex degree by means of the isomorphism of each regular the functional assigns . ${\displaystyle C_{c}(X)}$ ${\displaystyle \mu }$ ${\displaystyle f\mapsto \textstyle \int f\mathrm {d} \mu }$ ${\displaystyle C_{0}(X)}$ ${\displaystyle C_{0}(X)}$ ${\displaystyle C_{c}(X)}$ ${\displaystyle C_{0}(X)}$ ${\displaystyle C_{c}(X)}$ ${\displaystyle C_{0}(X)}$ ${\displaystyle X}$ ${\displaystyle C_{0}(X)^{\prime }}$ ${\displaystyle C_{0}(X)}$ ${\displaystyle C_{0}(X)^{\prime }}$ ${\displaystyle C_{0}(X)^{\prime }}$ ${\displaystyle \mu }$ ${\displaystyle f\mapsto \textstyle \int f\mathrm {d} \mu }$

See also

• Schwartz room
• Test function

Individual evidence

1. ↑ Alexander Grothendieck : Topological vector spaces . Gordon & Breach, 1973, ISBN 0-677-30025-5 , pp. 29 .
2. ^ Gerald B. Folland : Real Analysis . Modern Techniques and Their Applications. 2nd Edition. John Wiley & Sons, New York 1999, ISBN 0-471-31716-0 , pp. 132 .
3. ↑ Chun-Yen Chou, Notes on the separability of -algebras ${\displaystyle C^{*}}$ , Taiwanese Journal of Mathematics, 2012
4. ↑ Stephen Willard: General Topology . Addison-Wesley , 1970, pp. 57 .
5. ↑ Folland, p. 221 ff.