Uniform spaces are special
topological spaces in
which the important metric
notions of uniform convergence
and completeness can be properly generalized
(along with many other concepts now known as
uniform properties).
Uniform spaces were first introduced in 1936 by
André Weil (1906-1998) who
had been instrumental in founding the
Bourbaki group the previous year.
Uniform spaces were originally developed by Bourbaki
and John Tukey (1915-2000)
who came up with the definition of
uniform covers.
A metric space is complete when any Cauchy sequence in it converges.
Completeness is fundamentally a metric property (the definition of completeness depends critically on the definition of a distance, or some substitute thereof). Even if two distances are defined on the same set which induce the same topology on that space, it's quite possible that one distance defines a complete space and the other one doesn't.
A metric space is compact if and only if it's complete and totally bounded.
A subset of a metric space is said to be totally bounded when it can be covered by finitely many balls of radius r, for any given radius r. In a Euclidean space of infinitely many dimensions, a bounded set (like a ball of unit radius) need not be totally bounded. Actually, a closed ball is compact only in a space of finitely many dimensions.
By definition, a topological property is preserved by any homeomorphism. This is not always the case for completeness. For example, R is complete and it's homeomorphic to the open interval ]0,1[ which is not. (HINT : A positive sequence that tends to 0 in [0,1] isn't convergent in ]0,1[ .)
A metrizable space is defined as a topological space homeomorphic to a metric space. Such a space is called complete-metrizable when at least one metric space homeomorphic to it is complete. That's a topological property (since it's clearly preserved by homeomorphisms) but it's difficult to characterize in practice.
Recall that the following definitions hold for a function f which maps one metric space (X1 ,d1 ) into another (X2 ,d2 ) :
f is continuous when:
" e > 0 , " x Î X1 , $ d > 0 , d1 (x,y) ≤ d Þ d2 ( f (x), f (y) ) ≤ e
f is uniformly continuous when:
" e > 0 , $ d > 0 , " x Î X1 , d1 (x,y) ≤ d Þ d2 ( f (x), f (y) ) ≤ e
The order of the quantifiers matters: In the first case, d can depend on x. In the second case, it cannot.
If X1 is complete, so is f ( X1 )
when f is uniformly continuous.
That may not be the case if f is merely continuous.
A function is said to be Cauchy-regular (or Cauchy-continuous) if it transforms any Cauchy sequence into another Cauchy sequence. Uniformly continuous functions are Cauchy-regular.
Theorem : Continuity on a compact set is always uniform.
Proof : To establish that in the case of metric spaces (where uniform continuity is defined as a above) let's consider any continuous function f.
For any e > 0, the continuity of f implies that, for any given x, there's a quantity dx such that:
d1 (x,y) ≤ dx Þ d2 ( f (x), f (y) ) ≤ ½ e
To any x in X1 we associate a particular open set:
U x = { y : d1 (x,y) < ½ dx }
The family formed by all of these is an open cover of X1 (HINT: x ÎUx ). As X1 is assumed to be compact. we can extract from that family a finite subcover, for which we use the folowwing notation:
U xi = { y : d1 (xi ,y) < ½ d xi } with i = 1,2,3,4 ... n
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If f is a real function of a real variable defined on the interval A and differentiable in the interior Å of A, then f is uniformly continuous on A iff its derivative f ' is bounded on Å.
Uniform continuity
|
Heine-Cantor theorem
|
Eduard Heine (1821-1881)
Cauchy-regular functions
Uniformly
Continuous Functions Preserve Cauchy Sequences (Math Stack Exchange, 2012年12月06日).
Uniform Continuity and Cauchy Sequences (13:41)
by Peyam Tabrizian (Dr. Peyam, 2021年08月09日).
Uniform Continuity and Compactness (10:00)
by Peyam Tabrizian (Dr. Peyam, 2021年08月11日).
Uniform Continuity and Derivatives (9:56)
by Peyam Tabrizian (Dr. Peyam, 2021年08月23日).
Topological structures can be too permissive while the metric structures of normed spaces can be too strict a requirement. Uniform spaces seem just right to capture essential fruitful aspects of space. Uniform spaces are to uniform continuity what topological spaces are to ordinary continuity.
A uniform space is complete when every Cauchy filter in it converges.
What made it possible to define completeness in a metric space is the existence of a family of relations (i.e., subsets of the cartesian product) dependent on a single positive parameter a:
Ua = { (x,y) | d(x,y) < a }
The triangular inequality for the distance d enables us to construct a relation V = Ua/2 which is. loosely speaking, at most half as wide as the relation U = Ua . The crucial aspect can be expressed as follows, in terms of the composition of relations (this simple exercise is left to the reader).
V o V Í U
This expression no longer involves any explicit reference to distances. The postulated existence of a sequence of relations based on this composition pattern will enable us to generalize the notion of Cauchy sequences and completeness without using the notion of a distance...
A subset F of a poset (P, ≤) is a filter when it's a nonempty downward-directed upperset, which is to say:
P is always a filter of itself. The other filters of P are called proper filters. An ultrafilter is a maximal proper filter.
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nLab : Uniform spaces
Wikipedia :
Uniform spaces
|
Filters
&
ultrafilter (1937, Henri Cartan)
An important innovation of Bourbaki before 1945
by Connes, Serres, Cartier and Dixmier.
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Nets
(Moore-Smith sequence, 1922)
|
The word "net" was coined by
John L. Kelley (1916-1999) in 1955.
Cauchy nets in a metric space
by Jeff (math.stackexchange. 2012年08月28日).
A quasi-uniform space is complete when every Cauchy filter in it converges to a point in the space.
Uniform spaces are called quasi-uniform if the inverse of an entourage isn't necessarily an entourage.
This proposition is a theorem in ZFC. In ZF, it turns out to be equivalent to the Axiom of dependent choice (DC), a weak form of the full Axiom of choice which is sufficient for conducting real analysis without implying the repugnant existence of non-measurable sets of reals.
For that reason, Henri Garnir (1921-1985) has proposed to adopt DC or, equivalently, the Baire Category theorem instead of AC among the axioms of Set theory. This allowed him to postulate that every set of reals is almost open, which makes every set of reals Lebesgue-measurable and dismisses the Banach-Tarski paradox.
Let's try topological characterizations of completeness to see how such attempts fail. For example, let's examine the following property:
Any decreasing sequence of nonempty closed sets has a nonempty intersection:
This would seem like a good candidate for a topological characterization of completeness until you realize that it's not even true for a noncompact complete space like R in which there are indeed nested collection of nonempty closed sets with an empty intersection. Example: Ai = [i,¥[.
For families of compact closed sets, the above characterization still fails for metric spaces of infinitely many dimensions (where closed balls are not compact).
All told, a topological space can only be said to be complete with respect to a specific distance compatible with its topology (two different distances may induce the same topology but the space can be complete with respect to one metric and not the other). Such a space is called either topologically complete or complete-metrizable. There is simply no easy way to characterize that property...
Fréchet space =
Locally convex vector space,
complete with respect to a translation-invariant metric.
Cantor's intersection theorem
A Banach space is a normed vector space which is complete (which is to say that every Cauchy sequence in it converges). The concept is named after the Polish mathematician Stefan Banach (1892-1945) who axiomatized the idea in his doctoral dissertation (1920) and made it popular through his 1931 foundational book on functional analysis, which was translated in French the next year (Théorie des opérations linéaires, 1932).
Arguably, Banach spaces are the main backdrop for modern analysis, the branch of mathematics which revolves around the very notion of limit (it would be hazardous to discuss limits in a space that's not complete).
The Riesz-Fischer theorem (1907) states that Lp is a Banach space.
Lp spaces and Banach spaces
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Espaces de Lebesgue Lp
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Riesz-Fischer theorem (1907)
Ernst Fischer (1875-1954)
|
Frigyes Riesz (1880-1956)
|
Stefan Banach (1892-1945)
In 1906, Maurice Fréchet had proposed the general notion of a metric space without an underlying vector structure.
He realized that the key results that make Banach spaces interesting could also be obtained for vector spaces that are complete with respect to a distance not associated with a norm. He thus investigated structures more general than Banach spaces, which are now called Fréchet spaces :
A Fréchet space is a locally-convex vector space which is complete with respect to a given translation-invariant metric.
Wikipedia : Locally convex vector space | Fréchet space
