Sequence of functions

A sequence of functions that converges to the natural logarithm (red) in the non-hatched area . In this special case it is an n -th partial sum of a power series, and n is the number of summands.

A sequence of functions is a sequence whose individual members are functions . Function sequences and their convergence properties are of great importance for all areas of analysis . Above all, it is examined in what sense the sequence converges, whether the limit function inherits properties of the sequence or whether limit value formations can be exchanged for function sequences. The most important examples include series of functions such as power series , Fourier series or Dirichlet series . Here one also speaks of function series .

definition

A (real) sequence of functions is a sequence of functions . More generally, the definition and target set can also be other sets, for example intervals ; however, they must be the same for all functions. ${\displaystyle f_{1},f_{2},f_{3},\ldots }$ ${\displaystyle f_{i}\colon \mathbb {R} \to \mathbb {R} }$

A sequence of functions can be abstract as an illustration

${\displaystyle f\colon D\times \mathbb {N} \to Z,\quad (x,n)\mapsto f_{n}(x)}$

can be defined for a definition set and a target set . If the natural numbers were not chosen as the index set , one speaks of a family of functions. ${\displaystyle D}$ ${\displaystyle Z}$

Examples

Exchanging limit value and integral sign

For the episode , with ${\displaystyle (f_{n})_{n\in \mathbb {N} }\;}$ ${\displaystyle f_{n}\colon [0,2]\to \mathbb {R} }$

${\displaystyle f_{n}(x)={\begin{cases}n^{2}x&0\leq x\leq 1/n\\2n-n^{2}x&1/n\leq x\leq 2/n\\0&x\geq 2/n\end{cases}}}$

applies to every fixed ${\displaystyle x}$

${\displaystyle \lim _{n\to \infty }f_{n}(x)=0}$,

it converges point by point to the null function . However, applies to everyone ${\displaystyle n\in \mathbb {N} }$

${\displaystyle \int _{0}^{2}f_{n}(x)\,\mathrm {d} x=1,}$

so

${\displaystyle \lim _{n\to \infty }\int _{0}^{2}f_{n}(x)\,\mathrm {d} x\neq \int _{0}^{2}\lim _{n\to \infty }f_{n}(x)\,\mathrm {d} x.}$

Point-by-point convergence is therefore not sufficient for limit value and integral sign to be interchanged; In order for this exchange to be allowed, a more stringent convergence behavior, typically uniform convergence , majorized convergence or monotonic convergence , is sufficient.

Power series

In analysis, sequences of functions often appear as sums of functions, i.e. as a series , especially as a power series or, more generally, as a Laurent series .

Fourier analysis and approximation theory

In approximation theory , it is investigated how well functions can be represented as limit values ​​of function sequences, whereby the quantitative estimation of the error is of particular interest. The function sequences usually appear as series of functions, i.e. as a sum . For example, Fourier series in the -sense converge to the function to be represented. Better approximations in the sense of uniform convergence are often obtained with series of Chebyshev polynomials . ${\displaystyle \textstyle \sum _{n=1}^{N}f_{n}(x)}$ ${\displaystyle L^{2}}$

Stochastics

In stochastics , a random variable is defined as a measurable function of a measurement space with a probability measure. Sequences of random variables are therefore special sequences of functions. Statistics such as B. the sample mean function sequences. Important convergence properties of these function sequences are z. B. the strong law of large numbers and the weak law of large numbers . ${\displaystyle X}$ ${\displaystyle X:\Omega \to \mathbb {R} }$ ${\displaystyle (\Omega ,\Sigma ,P)}$ ${\displaystyle P(\Omega )=1}$ ${\displaystyle X_{n}}$ ${\displaystyle \textstyle {\bar {X}}_{N}:={\frac {1}{N}}\sum _{n=1}^{N}X_{n}}$

numerical Mathematics

In the numerical analysis dive function sequences, for example, in the solution of partial differential equations , wherein a (not necessarily linear) differential operator and the requested function. With the numerical solution, for example with the finite element method , functions are obtained as the solution of the discretized version of the equation , where the refinement denotes the discretization . When analyzing the numerical algorithm , the properties of the discretized solutions that form a sequence of functions are examined; In particular, it makes sense that the sequence of the discretized solutions converges towards the solution of the initial problem when the discretization is refined. ${\displaystyle \mathrm {D} f=0}$ ${\displaystyle \mathrm {D} }$ ${\displaystyle f}$ ${\displaystyle f_{n}}$ ${\displaystyle \mathrm {D} _{n}f=0}$ ${\displaystyle n}$ ${\displaystyle f_{n}}$ ${\displaystyle f_{n}}$

properties

monotony

→ Main article : Monotonous sequence of functions

A sequence of functions is called monotonically increasing (monotonically decreasing) if ( ) is for all . It is called monotonous if it is either monotonically decreasing or monotonically increasing. ${\displaystyle (f_{i})_{i\in \mathbb {N} }}$ ${\displaystyle D}$ ${\displaystyle f_{i}(x)\leq f_{i+1}(x)}$ ${\displaystyle f_{i}(x)\geq f_{i+1}(x)}$ ${\displaystyle x\in D}$

Pointwise limitation

A sequence of functions on a set , the set of values ​​of which is a normalized space, is called point-wise restricted if the set is restricted for each point . This set is the set of all values ​​that is assumed at that point by a function of the sequence. ${\displaystyle (f_{i})_{i\in \mathbb {N} }}$ ${\displaystyle D}$ ${\displaystyle x\in D}$ ${\displaystyle \{f_{i}(x)\mid i\in \mathbb {N} \}}$ ${\displaystyle x}$

Uniform limitation

A sequence of functions is bounded uniformly on a set , if a constant exists, so that for all and all . ${\displaystyle f_{i}\colon D\to \mathbb {R} ;i\in \mathbb {N} }$ ${\displaystyle A\subset D}$ ${\displaystyle c\in \mathbb {R} }$ ${\displaystyle \left|f_{i}(x)\right|\leq c}$ ${\displaystyle i\in \mathbb {N} }$ ${\displaystyle x\in A}$

A sequence of functions can only be uniformly restricted if every single function of the sequence is restricted. The supreme norm therefore exists for each individual function . A sequence of functions is then uniformly restricted if and only if it is restricted as a set of functions with respect to the supremum norm. ${\displaystyle f_{i}}$ ${\displaystyle \|f_{i}\|_{\infty }=\sup\{|f_{i}(x)|:x\in X\}}$

This is generalized to vector-valued functions : There is an arbitrary set, a real or complex normalized space with the norm . The set of functions defined on which are restricted with respect to the norm in is called and a norm is introduced on with which in turn makes a normalized space. Then a function sequence with defined functions is uniformly restricted if and only if the sequence is a subset of and is restricted as a subset of . ${\displaystyle D}$ ${\displaystyle Z}$ ${\displaystyle \|\cdot \|_{Z}\colon Z\to \mathbb {R} ^{+}}$ ${\displaystyle D}$ ${\displaystyle Z}$ ${\displaystyle B(D)}$ ${\displaystyle B(D)}$ ${\displaystyle \|f\|_{\infty }:=\sup\{\|f(x)\|_{Z}:x\in D\}}$ ${\displaystyle B(D)}$ ${\displaystyle D}$ ${\displaystyle B(D)}$ ${\displaystyle (B(D),\|\cdot \|_{\infty })}$

An evenly restricted sequence of functions is necessarily also restricted point by point.

Locally uniform limitation

A sequence of functions is locally evenly restricted to an open set , if an open environment and a constant exist for each , so that applies to all and all . ${\displaystyle f_{i}\colon D\to \mathbb {R} ;i\in \mathbb {N} }$ ${\displaystyle A\subset D}$ ${\displaystyle x_{0}\in A}$ ${\displaystyle U(x_{0})}$ ${\displaystyle c\in \mathbb {R} }$ ${\displaystyle \left|f_{i}(x)\right|\leq c}$ ${\displaystyle i\in \mathbb {N} }$ ${\displaystyle x\in U(x_{0})}$

Convergence terms

The limit value of a function sequence is called the limit function . Since the function sequences occurring in the applications can behave very differently as the index increases, it is necessary to introduce a large number of different convergence terms for function sequences. From a more abstract point of view it is mostly a question of the convergence with regard to certain norms or general topologies on the corresponding function spaces ; however, other concepts of convergence appear occasionally. ${\displaystyle f}$

The various concepts of convergence differ mainly in the implied properties of the limit function. The most important are:

Classic convergence terms

Point-by-point convergence

Does the point-by-point limit exist

${\displaystyle f(x)=\lim _{n\to \infty }f_{n}(x)}$

at every point of the domain, the sequence of functions is called point-wise convergent. For example ${\displaystyle x}$

${\displaystyle \lim _{n\to \infty }\cos ^{2n}x={\begin{cases}1&x=\pi k,\ k\in \mathbb {Z} \\0&\mathrm {sonst} ,\end{cases}}}$

the limit function is therefore discontinuous.

Uniform convergence

A sequence of functions is uniformly convergent to a function if the maximum differences between and converge to zero. This concept of convergence is convergence in the sense of the supremacy norm . ${\displaystyle (f_{n})_{n}}$ ${\displaystyle f}$ ${\displaystyle f_{n}}$ ${\displaystyle f}$

Uniform convergence implies some properties of the limit function if the sequence members have them:

• The uniform limit of continuous functions is continuous.
• The uniform limit of a sequence (Riemann or Lebesgue) integrable functions on a compact interval is (Riemann or Lebesgue) integrable, and the integral of the limit function is the limit of the integrals of the sequence members: If uniformly converges to , then the following applies ${\displaystyle (f_{n})_{n}}$ ${\displaystyle f}$

${\displaystyle \lim _{n\to \infty }\int _{a}^{b}f_{n}=\int _{a}^{b}f.}$

• If a sequence of differentiable functions converges point by point to a function and if the sequence of the derivatives converges uniformly, then it is differentiable and it applies ${\displaystyle (f_{n})_{n}}$ ${\displaystyle f}$ ${\displaystyle f}$

${\displaystyle \lim _{n\to \infty }f_{n}'=f'.}$

Locally uniform convergence

Many series in function theory , especially power series , do not converge uniformly because the convergence gets worse and worse for increasing arguments. If one demands uniform convergence only locally, that is, in the vicinity of each point, one arrives at the concept of locally uniform convergence , which is sufficient for many applications in analysis. As in the case of uniform convergence, the continuity of the sequence members is also transferred to the limit function in the case of locally uniform convergence.

Compact convergence

A similarly good concept of convergence is that of compact convergence , which only requires uniform convergence on compact subsets. The compact convergence follows from the locally uniform convergence; for locally compact spaces , which often occur in applications, the reverse applies.

Normal convergence

In mathematics, the concept of normal convergence is used to characterize infinite series of functions. The term was introduced by the French mathematician René Louis Baire .

Dimension-theoretical convergence terms

In the case of the convergence concepts of the theory of measure , the limit function is usually not unambiguous, but only clearly defined almost everywhere. Alternatively, this convergence can also be understood as the convergence of equivalence classes of functions that agree almost everywhere. The limit value is then clearly defined as such an equivalence class.

Point convergence almost everywhere

→ Main article : Point-wise convergence μ-almost everywhere

If a measurement space and a sequence of measurable functions with a definition set are given, the sequence of functions is called point-wise convergent almost everywhere with regard to the point-wise limit value ${\displaystyle (\Omega ,\Sigma ,\mu )}$ ${\displaystyle f_{n}}$ ${\displaystyle \Omega }$ ${\displaystyle \mu }$

${\displaystyle f(x)=\lim _{n\to \infty }f_{n}(x)}$

exists almost everywhere with respect , i.e. if a set of measure zero ( ) exists, so that restricted to the complement it converges point by point. ${\displaystyle \mu }$ ${\displaystyle Z\in \Sigma }$ ${\displaystyle \mu (Z)=0}$ ${\displaystyle f_{n}}$ ${\displaystyle \Omega \backslash Z}$

The convergence almost everywhere with regard to a probability measure is called almost certain convergence in stochastics .

For example

${\displaystyle \lim _{n\to \infty }\cos ^{2n}x=0}$point by point almost everywhere with regard to the Lebesgue measure .

Another example is the sequence of functions , where for , ${\displaystyle f_{n}:[0,1]\to [0,1]}$ ${\displaystyle n=2^{r}+s\;}$ ${\displaystyle 0\leq s\leq 2^{r}-1}$

${\displaystyle f_{2^{r}+s}(x):={\begin{cases}1&{\frac {s}{2^{r}}}\leq x\leq {\frac {s+1}{2^{r}}}\\0&\mathrm {sonst.} \end{cases}}}$

This sequence does not converge for any , since it assumes the values ​​0 and 1 infinitely often for every fixed one. For each subsequence , however, a subsequence can be specified so that ${\displaystyle x\in [0,1]}$ ${\displaystyle x}$ ${\displaystyle f_{n_{k}},k\in \mathbb {N} }$ ${\displaystyle f_{n_{k_{l}}},l\in \mathbb {N} }$

${\displaystyle \lim _{l\to \infty }f_{n_{k_{l}}}(x)=0}$ point by point almost everywhere with regard to the Lebesgue measure.

If there were a topology of point-wise convergence almost everywhere, it would follow that every subsequence of contains a subsequence that converges to 0, that must converge to 0. However, since it does not converge, there cannot be a topology of convergence almost everywhere. The point-by-point convergence almost everywhere is thus an example of a concept of convergence which, although satisfying the Fréchet axioms , cannot be generated by a topology. ${\displaystyle f_{n}}$ ${\displaystyle f_{n}}$ ${\displaystyle f_{n}}$

Convergence by measure

→ Main articles : Convergence made to measure  and local convergence made to measure

In a measure space , a sequence of functions measurable on it is called convergent in measure to a function , if for each ${\displaystyle (\Omega ,\Sigma ,\mu )}$ ${\displaystyle f_{n}}$ ${\displaystyle f}$ ${\displaystyle \varepsilon >0}$

${\displaystyle \lim _{n\to \infty }\mu \left(\{x:\;|f_{n}(x)-f(x)|\geq \varepsilon \}\right)=0}$

applies.

In a finite measure space, i.e. if it is true, the convergence is weaker in terms of measure than the convergence almost everywhere: If a sequence of measurable functions converges almost everywhere on the function , then it also converges on the measure against . ${\displaystyle \mu (\Omega )<\infty }$ ${\displaystyle f_{n}}$ ${\displaystyle f}$ ${\displaystyle f}$

In stochastics, the degree of convergence is called stochastic convergence or convergence in probability .

A weakening of the convergence by measure is the convergence locally by measure . Both terms agree on finite measure spaces .

L ^p -convergence and convergence in Sobolev spaces

→ Main article : Convergence in the p-th mean

A sequence of functions is called convergent to or convergent in the p-th mean if it converges in the sense of the corresponding L ^p -space, i.e. if ${\displaystyle f_{n}}$ ${\displaystyle L^{p}}$ ${\displaystyle f}$ ${\displaystyle {\mathcal {L}}^{p}(\Omega ,{\mathcal {A}},\mu ;E)}$

${\displaystyle \lim _{n\to \infty }\|f_{n}-f\|_{p}=\lim _{n\to \infty }\left(\int _{\Omega }\|f_{n}(x)-f(x)\|^{p}\,\mathrm {d} \mu (x)\right)^{1/p}=0.}$

If it is a finite measure, then it follows from the inequality of the generalized mean values that a constant exists such that ; In particular, it then follows from the convergence of against also the convergence of against . ${\displaystyle \mu }$ ${\displaystyle \mu (\Omega )<\infty }$ ${\displaystyle q\geq p\geq 0}$ ${\displaystyle k\in \mathbb {R} ^{+}}$ ${\displaystyle \|f\|_{p}\leq k\|f\|_{q}}$ ${\displaystyle L^{q}}$ ${\displaystyle f_{n}}$ ${\displaystyle f}$ ${\displaystyle L^{p}}$ ${\displaystyle f_{n}}$ ${\displaystyle f}$

The convergence follows from the convergence according to the extent to which one can derive from the Chebyshev inequality in the form ${\displaystyle L^{p}}$

${\displaystyle \mu \{x:|f_{n}(x)-f(x)|\geq \varepsilon \}\leq {\frac {1}{\varepsilon ^{p}}}\int _{\Omega }|f_{n}(x)-f(x)|^{p}{\rm {d}}\mu (x)}$

sees.

A generalization of the L ^p convergence is the convergence in Sobolew spaces , which not only takes into account the convergence of the function values, but also the convergence of certain derivatives . Sobolew's embedding theorem describes the dependencies of the convergence concepts in the different Sobolew spaces.

Almost uniform convergence

→ Main article : Almost uniform convergence

In a measure space , a sequence of real or complex-valued functions measurable on it is called almost uniformly convergent to a function if for each there is a set such that and converges uniformly to on the complement . ${\displaystyle (\Omega ,\Sigma ,\mu )}$ ${\displaystyle f_{n}}$ ${\displaystyle f}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle A\in \Sigma }$ ${\displaystyle \mu (A)<\varepsilon }$ ${\displaystyle f_{n}}$ ${\displaystyle \Omega \backslash A}$ ${\displaystyle f}$

From the almost uniform convergence, point-by-point convergence follows almost everywhere; from the set of Jegorow follows that in a finite dimensional space, conversely, almost everywhere follows from the pointwise convergence almost uniform convergence. In a finite measure space, especially for real-valued random variables, convergence almost everywhere and almost uniform convergence of real-valued function sequences are equivalent.

The convergence also follows from the almost uniform convergence in proportion. Conversely, a sequence that is convergent in terms of measure contains a subsequence that converges almost uniformly (and therefore almost everywhere) to the same limit sequence.

Uniform convergence almost everywhere

→ Main article : Uniform convergence μ-almost everywhere

In a measure space , a sequence of real or complex-valued functions that can be measured on it is called almost everywhere uniformly convergent to a function if there is a null set such that it converges uniformly to the complement . For sequences of limited functions this is essentially the convergence in space . Uniform convergence almost everywhere can easily be confused with almost uniform convergence because of the very similar designation, as Paul Halmos criticizes in his textbook on measurement theory. ${\displaystyle (\Omega ,\Sigma ,\mu )}$ ${\displaystyle f_{n}}$ ${\displaystyle f}$ ${\displaystyle Z\in \Sigma }$ ${\displaystyle f_{n}}$ ${\displaystyle \Omega \backslash Z}$ ${\displaystyle f}$ ${\displaystyle L^{\infty }(\Omega ,\Sigma ,\mu )}$

Weak convergence

→ Main article : Weak convergence in Lp

The weak convergence for function sequences is a special case of weak convergence in the sense of functional analysis , which is generally defined for normalized spaces . Note that there are several different concepts of weak convergence in functional analysis , measure theory, and stochastics that should not be confused with one another.

For is called a sequence of functions consisting of weakly convergent against if it holds for all that ${\displaystyle p\in [1,\infty )}$ ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle {\mathcal {L}}^{p}}$ ${\displaystyle f}$ ${\displaystyle g\in {\mathcal {L}}^{q}}$

${\displaystyle \lim _{n\to \infty }\int _{X}f_{n}g\mathrm {d} \mu =\int _{X}fg\mathrm {d} \mu }$

is. It is by defined. ${\displaystyle q}$ ${\displaystyle {\frac {1}{p}}+{\frac {1}{q}}=1}$

Overview of the mass-theoretical types of convergence

An overview of the types of convergence based on the theory of dimensions

The overview opposite is taken from the textbook Introduction to Measurement Theory by Ernst Henze , who in turn refers to older predecessors. It clarifies the logical relationships between the types of convergence for a sequence of measurable functions on a measurement space . A black, solid arrow means that the type of convergence at the arrowhead follows the type of convergence at the origin of the arrow. This only applies to the blue dashed arrows if it is required. The implication applies to the red dash-dot arrows if the sequence is restricted by an integrable function. ${\displaystyle (\Omega ,\Sigma ,\mu )}$ ${\displaystyle \mu (\Omega )<\infty }$ ${\displaystyle \mu }$

Hierarchical order Concepts of convergence in spaces with finite dimensions

In dimensional spaces with finite dimensions , if so , it is largely possible to order the different concepts of convergence according to their strength. This is especially true in probability spaces , since yes there . ${\displaystyle (\Omega ,\Sigma ,\mu )}$ ${\displaystyle \mu (\Omega )<\infty }$ ${\displaystyle \mu (\Omega )=1}$

From the uniform convergence, the convergence follows the measure in two different ways, one of which leads via point-wise convergence:

• ${\displaystyle f_{n}\to f}$uniformly locally uniformly (ie uniformly on a neighborhood of each point). ${\displaystyle \Rightarrow f_{n}\to f}$
• ${\displaystyle f_{n}\to f}$locally uniformly compact (ie uniformly on each compact subset). ${\displaystyle \Rightarrow f_{n}\to f}$
• ${\displaystyle f_{n}\to f}$compact point by point (every single point is a compact subset). ${\displaystyle \Rightarrow f_{n}\to f}$
• ${\displaystyle f_{n}\to f}$point by point almost everywhere (or almost certainly). ${\displaystyle \Rightarrow f_{n}\to f}$
• ${\displaystyle f_{n}\to f}$pointwise almost everywhere almost evenly. ${\displaystyle \Leftrightarrow f_{n}\to f}$
• ${\displaystyle f_{n}\to f}$almost uniformly in terms of measure (or stochastically or in probability). ${\displaystyle \Rightarrow f_{n}\to f}$

The other path from uniform convergence to convergence in terms of measure is via convergence: ${\displaystyle L^{p}}$

• ${\displaystyle f_{n}\to f}$evenly in . ${\displaystyle \Rightarrow f_{n}\to f}$ ${\displaystyle L^{\infty }}$
• ${\displaystyle f_{n}\to f}$in in for all real ones . ${\displaystyle L^{\infty }}$ ${\displaystyle \Rightarrow f_{n}\to f}$ ${\displaystyle L^{p}}$ ${\displaystyle 0<p<\infty }$
• ${\displaystyle f_{n}\to f}$in in for all real ones . ${\displaystyle L^{p}}$ ${\displaystyle \Rightarrow f_{n}\to f}$ ${\displaystyle L^{q}}$ ${\displaystyle 0<q<p}$
• ${\displaystyle f_{n}\to f}$in for the measure (or stochastically or in probability). ${\displaystyle L^{p}}$ ${\displaystyle 0<p\leq \infty \quad \Rightarrow f_{n}\to f}$

From the convergence according to measure one arrives at the weak convergence:

• ${\displaystyle f_{n}\to f}$weak (or in distribution) according to measure . ${\displaystyle \Rightarrow f_{n}\to f}$

Important theorems about function sequences

• Arzelà-Ascoli's theorem
• Dini's theorem
• Yegorov's theorem

literature

• Heinz Bauer : Measure and integration theory. 2nd Edition. De Gruyter, Berlin 1992, ISBN 3-11-013626-0 (hardcover), ISBN 3-11-013625-2 (paperback), from p. 91 (§15 Convergence Sentences) and from p. 128 (§20 Stochastic Convergence) .
• Jürgen Elstrodt : Measure and Integration Theory 4th edition. Springer, Berlin 2005, ISBN 3-540-21390-2 , (Describes in detail the relationships between the different types of convergence).

Individual evidence

1. ↑ J. Cigler, H.-C. Reichel: topology. A basic lecture. Bibliographisches Institut, Mannheim 1978. ISBN 3-411-00121-6 . P. 88, exercise 6
2. ↑ AN Kolmogorow and SV Fomin: Real functions and functional analysis. Deutscher Verlag der Wissenschaften, Berlin 1975, 5.4.6, definition 4.
3. ↑ AN Kolmogorow and SV Fomin: Real functions and functional analysis. Deutscher Verlag der Wissenschaften, Berlin 1975, 5.4.6, sentence 7.
4. ^ Marek Fisz : Probability calculation and mathematical statistics. Deutscher Verlag der Wissenschaften, Berlin 1989, p. 212.
5. ^ Robert B. Ash: Real Analysis and Probability. Academic Press, New York 1972. ISBN 0-12-065201-3 . Theorem 2.5.1.
6. ^ Robert B. Ash: Real Analysis and Probability. Academic Press, New York 1972. ISBN 0-12-065201-3 . P. 93.
7. ^ ^{A b} Robert B. Ash: Real Analysis and Probability. Academic Press, New York 1972. ISBN 0-12-065201-3 . Theorem 2.5.2.
8. ^ Robert B. Ash: Real Analysis and Probability. Academic Press, New York 1972. ISBN 0-12-065201-3 . Theorem 2.5.5.
9. ^ Robert B. Ash: Real Analysis and Probability. Academic Press, New York 1972. ISBN 0-12-065201-3 . Theorem 2.5.3.
10. ^ Paul Halmos: Measure Theory , Springer-Verlag, Graduate Texts in Mathematics, ISBN 978-1-4684-9442-6 , §22, page 90
11. ↑ Ernst Henze: Introduction to Dimension Theory , BI, Mannheim, 1971, ISBN 3-411-03102-6 , Chapter 4.6, page 146