Heine-Borel Theorem - Revision history

Lila at 19:36, 27 October 2021

2021-10-27T19:36:53Z

Lila at 19:35, 27 October 2021

2021-10-27T19:35:07Z

Lila: Created page with "In real analysis the '''Heine–Borel theorem''', named after Eduard Heine and Émile Borel, states: For a subset ''S'' of Euclidean space '''R'''^''n..."

2021-10-25T20:19:47Z

Created page with "In real analysis the '''Heine–Borel theorem''', named after Eduard Heine and Émile Borel, states: For a subset ''S'' of Euclidean space '''R'''''n..."

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In [[real analysis]] the '''Heine–Borel theorem''', named after [[Eduard Heine]] and [[Émile Borel]], states:

For a [[subset]] ''S'' of [[Euclidean space]] '''R'''''n'', the following two statements are equivalent:
*''S'' is [[closed set|closed]] and [[bounded set|bounded]]
*''S'' is [[compact space|compact]], that is, every open [[cover (topology)|cover]] of ''S'' has a finite subcover.

==History and motivation==

The history of what today is called the Heine–Borel theorem starts in the 19th century, with the search for solid foundations of real analysis. Central to the theory was the concept of [[uniform continuity]] and the theorem stating that every [[continuous function]] on a closed interval is uniformly continuous. [[Peter Gustav Lejeune Dirichlet]] was the first to prove this and implicitly he used the existence of a finite subcover of a given open cover of a closed interval in his proof.<ref name="Sundström"/> He used this proof in his 1852 lectures, which were published only in 1904.<ref name="Sundström">{{cite journal | journal = [[American Mathematical Monthly]] | title = A Pedagogical History of Compactness | last1 = Raman-Sundström | first1 = Manya | date = August–September 2015 | volume = 122 | issue = 7 | pages = 619–635 | jstor = 10.4169/amer.math.monthly.122.7.619| doi = 10.4169/amer.math.monthly.122.7.619 | arxiv = 1006.4131 | s2cid = 119936587 }}</ref> Later [[Eduard Heine]], [[Karl Weierstrass]] and [[Salvatore Pincherle]] used similar techniques. [[Émile Borel]] in 1895 was the first to state and prove a form of what is now called the Heine–Borel theorem. His formulation was restricted to [[countable set|countable]] covers. Pierre Cousin (1895), [[Lebesgue]] (1898) and [[Arthur Schoenflies|Schoenflies]] (1900) generalized it to arbitrary covers.<ref name="sundstrom_2010">{{cite arXiv |last=Sundström |first=Manya Raman | eprint=1006.4131v1 |title=A pedagogical history of compactness |class=math.HO |year=2010 }}</ref>

== Proof ==

'''If a set is compact, then it must be closed.'''

Let ''S'' be a subset of '''R'''''n''. Observe first the following: if ''a'' is a [[limit point]] of ''S'', then any finite collection ''C'' of open sets, such that each open set ''U'' ∈ ''C'' is disjoint from some [[Neighbourhood (mathematics)#Definition|neighborhood]] ''V''''U'' of ''a'', fails to be a cover of ''S''. Indeed, the intersection of the finite family of sets ''V''''U'' is a neighborhood ''W'' of ''a'' in '''R'''''n''. Since ''a'' is a limit point of ''S'', ''W'' must contain a point ''x'' in ''S''. This ''x'' ∈ ''S'' is not covered by the family ''C'', because every ''U'' in ''C'' is disjoint from ''V''''U'' and hence disjoint from ''W'', which contains ''x''.

If ''S'' is compact but not closed, then it has a limit point ''a'' not in ''S''. Consider a collection {{nowrap|''C'' ′}} consisting of an open neighborhood ''N''(''x'') for each ''x'' ∈ ''S'', chosen small enough to not intersect some neighborhood ''V''''x'' of ''a''. Then {{nowrap|''C'' ′}} is an open cover of ''S'', but any finite subcollection of {{nowrap|''C'' ′}} has the form of ''C'' discussed previously, and thus cannot be an open subcover of ''S''. This contradicts the compactness of ''S''. Hence, every limit point of ''S'' is in ''S'', so ''S'' is closed.

The proof above applies with almost no change to showing that any compact subset ''S'' of a [[Hausdorff space|Hausdorff]] topological space ''X'' is closed in ''X''.

'''If a set is compact, then it is bounded.'''

Let <math>S</math> be a compact set in <math>\mathbf{R}^n</math>, and <math>U_x</math> a ball of radius 1 centered at <math>x\in\mathbf{R}^n</math>. Then the set of all such balls centered at <math>x\in S</math> is clearly an open cover of <math>S</math>, since <math>\cup_{x\in S} U_x</math> contains all of <math>S</math>. Since <math>S</math> is compact, take a finite subcover of this cover. This subcover is the finite union of balls of radius 1. Consider all pairs of centers of these (finitely many) balls (of radius 1) and let <math>M</math> be the maximum of the distances between them. Then if <math>C_p</math> and <math>C_q</math> are the centers (respectively) of unit balls containing arbitrary <math>p,q\in S</math>, the triangle inequality says:
:<math>
d(p, q)\le d(p, C_p) + d(C_p, C_q) + d(C_q, q)\le 1 + M + 1 = M + 2.
</math>
So the diameter of <math>S</math> is bounded by <math>M+2</math>.

'''A closed subset of a compact set is compact.'''

Let ''K'' be a closed subset of a compact set ''T'' in '''R'''''n'' and let ''C''''K'' be an open cover of ''K''. Then {{nowrap|''U'' {{=}} '''R'''''n'' \ ''K''}} is an open set and

:<math> C_T = C_K \cup \{U\} </math>

is an open cover of ''T''. Since ''T'' is compact, then ''C''''T'' has a finite subcover <math> C_T',</math> that also covers the smaller set ''K''. Since ''U'' does not contain any point of ''K'', the set ''K'' is already covered by <math> C_K' = C_T' \setminus \{U\}, </math> that is a finite subcollection of the original collection ''C''''K''. It is thus possible to extract from any open cover ''C''''K'' of ''K'' a finite subcover.

'''If a set is closed and bounded, then it is compact.'''

If a set ''S'' in '''R'''''n'' is bounded, then it can be enclosed within an ''n''-box

: <math> T_0 = [-a, a]^n</math>

where ''a'' > 0. By the property above, it is enough to show that ''T''0 is compact.

Assume, by way of contradiction, that ''T''0 is not compact. Then there exists an infinite open cover ''C'' of ''T''0 that does not admit any finite subcover. Through bisection of each of the sides of ''T''0, the box ''T''0 can be broken up into 2''n'' sub ''n''-boxes, each of which has diameter equal to half the diameter of ''T''0. Then at least one of the 2''n'' sections of ''T''0 must require an infinite subcover of ''C'', otherwise ''C'' itself would have a finite subcover, by uniting together the finite covers of the sections. Call this section ''T''1.

Likewise, the sides of ''T''1 can be bisected, yielding 2''n'' sections of ''T''1, at least one of which must require an infinite subcover of ''C''. Continuing in like manner yields a decreasing sequence of nested ''n''-boxes:

:<math> T_0 \supset T_1 \supset T_2 \supset \ldots \supset T_k \supset \ldots </math>

where the side length of ''T''''k'' is {{nowrap|(2 ''a'') / 2''k''}}, which tends to 0 as ''k'' tends to infinity. Let us define a sequence (''x''k) such that each ''x''k is in ''T''k. This sequence is Cauchy, so it must converge to some limit ''L''. Since each ''T''''k'' is closed, and for each ''k'' the sequence (''x''k) is eventually always inside ''T''k, we see that ''L'' ∈ ''T''k for each ''k''.

Since ''C'' covers ''T''0, then it has some member ''U'' ∈ ''C'' such that ''L'' ∈ ''U''. Since ''U'' is open, there is an ''n''-ball {{nowrap|''B''(''L'') ⊆ ''U''}}. For large enough ''k'', one has {{nowrap|''T''''k'' ⊆ ''B''(''L'') ⊆ ''U''}}, but then the infinite number of members of ''C'' needed to cover ''Tk'' can be replaced by just one: ''U'', a contradiction.

Thus, ''T''0 is compact. Since ''S'' is closed and a subset of the compact set ''T''0, then ''S'' is also compact (see above).

== Heine–Borel property ==
The Heine–Borel theorem does not hold as stated for general [[Metric space|metric]] and [[topological vector space]]s, and this gives rise to the necessity to consider special classes of spaces where this proposition is true. They are called the '''spaces with the Heine–Borel property'''.

===In the theory of metric spaces===
A [[metric space]] <math>(X,d)</math> is said to have the '''Heine–Borel property''' if each closed bounded<ref>A set <math>B</math> in a metric space <math>(X,d)</math> is said to be ''bounded'' if it is contained in a ball of a finite radius, i.e. there exists <math>x\in X</math> and <math>r>0</math> such that <math>B\subseteq\{x\in X: \ d(x,a)\le r\}</math>.</ref> set in <math>X</math> is compact.

Many metric spaces fail to have the Heine–Borel property, such as the metric space of [[rational number]]s (or indeed any incomplete metric space). Complete metric spaces may also fail to have the property; for instance, no infinite-dimensional [[Banach space]]s have the Heine–Borel property (as metric spaces). Even more trivially, if the real line is not endowed with the usual metric, it may fail to have the Heine–Borel property.

A metric space <math>(X,d)</math> has a Heine–Borel metric which is Cauchy locally identical to <math>d</math> if and only if it is [[complete space|complete]], [[sigma-compact|<math>\sigma</math>-compact]], and [[locally compact space|locally compact]].{{sfn|Williamson|Janos|1987}}

===In the theory of topological vector spaces===
A [[topological vector space]] <math>X</math> is said to have the '''Heine–Borel property'''{{sfn|Kirillov|Gvishiani|1982|loc=Theorem 28}} (R.E. Edwards uses the term ''boundedly compact space''{{sfn|Edwards|1965|loc=8.4.7}}) if each closed bounded<ref>A set <math>B</math> in a topological vector space <math>X</math> is said to be ''bounded'' if for each neighborhood of zero <math>U</math> in <math>X</math> there exists a scalar <math>\lambda</math> such that <math>B\subseteq\lambda\cdot U</math>.</ref> set in <math>X</math> is compact.<ref>In the case when the topology of a topological vector space <math>X</math> is generated by some metric <math>d</math> this definition is not equivalent to the definition of the Heine–Borel property of <math>X</math> as a metric space, since the notion of bounded set in <math>X</math> as a metric space is different from the notion of bounded set in <math>X</math> as a topological vector space. For instance, the space <math>{\mathcal C}^\infty[0,1]</math> of smooth functions on the interval <math>[0,1]</math> with the metric <math>d(x,y)=\sum_{k=0}^\infty\frac{1}{2^k}\cdot\frac{\max_{t\in[0,1]}|x^{(k)}(t)-y^{(k)}(t)|}{1+\max_{t\in[0,1]}|x^{(k)}(t)-y^{(k)}(t)|}</math> (here <math>x^{(k)}</math> is the <math>k</math>-th derivative of the function <math>x\in {\mathcal C}^\infty[0,1]</math>) has the Heine–Borel property as a topological vector space but not as a metric space.</ref> No infinite-dimensional [[Banach space]]s have the Heine–Borel property (as topological vector spaces). But some infinite-dimensional [[Fréchet space]]s do have, for instance, the space <math>C^\infty(\Omega)</math> of smooth functions on an open set <math>\Omega\subset\mathbb{R}^n</math>{{sfn|Edwards|1965|loc=8.4.7}} and the space <math>H(\Omega)</math> of holomorphic functions on an open set <math>\Omega\subset\mathbb{C}^n</math>.{{sfn|Edwards|1965|loc=8.4.7}} More generally, any quasi-complete [[nuclear space]] has the Heine–Borel property. All [[Montel space]]s have the Heine–Borel property as well.

==Licensing==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Heine%E2%80%93Borel_theorem Heine-Borel theorem, Wikipedia] under a CC BY-SA license

@@ Line 29: / Line 29: @@
 '''A closed subset of a compact set is compact.'''
-Let ''K'' be a closed subset of a compact set ''T'' in '''R'''<sup>''n''</sup> and let ''C''<sub>''K''</sub> be an open cover of ''K''.  Then {{nowrap|''U'' {{=}} '''R'''<sup>''n''</sup> \ ''K''}} is an open set and
+Let ''K'' be a closed subset of a compact set ''T'' in '''R'''<sup>''n''</sup> and let ''C''<sub>''K''</sub> be an open cover of ''K''.  Then ''U'' = '''R'''<sup>''n''</sup> \ ''K'' is an open set and
 :<math> C_T = C_K \cup \{U\} </math>
@@ Line 49: / Line 49: @@
 :<math> T_0 \supset T_1 \supset T_2 \supset \ldots \supset T_k \supset \ldots </math>
-where the side length of ''T''<sub>''k''</sub> is {{nowrap|(2&thinsp;''a'')&thinsp;/&thinsp;2<sup>''k''</sup>}}, which tends to 0 as ''k'' tends to infinity. Let us define a sequence (''x''<sub>k</sub>) such that each ''x''<sub>k</sub> is in ''T''<sub>k</sub>. This sequence is Cauchy, so it must converge to some limit ''L''. Since each ''T''<sub>''k''</sub> is closed, and for each ''k'' the sequence  (''x''<sub>k</sub>) is eventually always inside ''T''<sub>k</sub>, we see that ''L''&nbsp;∈&nbsp;''T''<sub>k</sub> for each ''k''.
+where the side length of ''T''<sub>''k''</sub> is (2&thinsp;''a'')&thinsp;/&thinsp;2<sup>''k''</sup>, which tends to 0 as ''k'' tends to infinity. Let us define a sequence (''x''<sub>k</sub>) such that each ''x''<sub>k</sub> is in ''T''<sub>k</sub>. This sequence is Cauchy, so it must converge to some limit ''L''. Since each ''T''<sub>''k''</sub> is closed, and for each ''k'' the sequence  (''x''<sub>k</sub>) is eventually always inside ''T''<sub>k</sub>, we see that ''L''&nbsp;∈&nbsp;''T''<sub>k</sub> for each ''k''.
-Since ''C'' covers ''T''<sub>0</sub>, then it has some member ''U''&nbsp;∈ ''C'' such that ''L''&nbsp;∈ ''U''.  Since ''U'' is open, there is an ''n''-ball {{nowrap|''B''(''L'') ⊆ ''U''}}.  For large enough ''k'', one has {{nowrap|''T''<sub>''k''</sub> ⊆ ''B''(''L'') ⊆ ''U''}}, but then the infinite number of members of ''C'' needed to cover ''T<sub>k</sub>'' can be replaced by just one: ''U'', a contradiction.
+Since ''C'' covers ''T''<sub>0</sub>, then it has some member ''U''&nbsp;∈ ''C'' such that ''L''&nbsp;∈ ''U''.  Since ''U'' is open, there is an ''n''-ball ''B''(''L'') ⊆ ''U''.  For large enough ''k'', one has ''T''<sub>''k''</sub> ⊆ ''B''(''L'') ⊆ ''U'', but then the infinite number of members of ''C'' needed to cover ''T<sub>k</sub>'' can be replaced by just one: ''U'', a contradiction.
 Thus, ''T''<sub>0</sub> is compact.  Since ''S'' is closed and a subset of the compact set ''T''<sub>0</sub>, then ''S'' is also compact (see above).

@@ Line 1: / Line 1: @@
-In [[real analysis]] the '''Heine–Borel theorem''', named after [[Eduard Heine]] and [[Émile Borel]], states:
+In real analysis the '''Heine–Borel theorem''', named after Eduard Heine and Émile Borel, states:
-For a [[subset]] ''S'' of [[Euclidean space]] '''R'''<sup>''n''</sup>, the following two statements are equivalent:
+For a subset ''S'' of Euclidean space '''R'''<sup>''n''</sup>, the following two statements are equivalent:
-*''S'' is [[closed set|closed]] and [[bounded set|bounded]]
+*''S'' is closed and bounded
-*''S'' is [[compact space|compact]], that is, every open [[cover (topology)|cover]] of ''S'' has a finite subcover.
+*''S'' is compact, that is, every open cover of ''S'' has a finite subcover.
 ==History and motivation==
-The history of what today is called the Heine–Borel theorem starts in the 19th century, with the search for solid foundations of real analysis. Central to the theory was the concept of [[uniform continuity]] and the theorem stating that every [[continuous function]] on a closed interval is uniformly continuous. [[Peter Gustav Lejeune Dirichlet]] was the first to prove this and implicitly he used the existence of a finite subcover of a given open cover of a closed interval in his proof.<ref name="Sundström"/> He used this proof in his 1852 lectures, which were published only in 1904.<ref name="Sundström">{{cite journal | journal = [[American Mathematical Monthly]] | title = A Pedagogical History of Compactness | last1 = Raman-Sundström | first1 = Manya | date = August–September 2015 | volume = 122 | issue = 7 | pages = 619–635 | jstor = 10.4169/amer.math.monthly.122.7.619| doi = 10.4169/amer.math.monthly.122.7.619 | arxiv = 1006.4131 | s2cid = 119936587 }}</ref> Later [[Eduard Heine]], [[Karl Weierstrass]] and [[Salvatore Pincherle]] used similar techniques. [[Émile Borel]] in 1895 was the first to state and prove a form of what is now called the Heine–Borel theorem. His formulation was restricted to [[countable set|countable]] covers. Pierre Cousin (1895), [[Lebesgue]] (1898) and [[Arthur Schoenflies|Schoenflies]] (1900) generalized it to arbitrary covers.<ref name="sundstrom_2010">{{cite arXiv |last=Sundström |first=Manya Raman | eprint=1006.4131v1 |title=A pedagogical history of compactness |class=math.HO |year=2010 }}</ref>
+The history of what today is called the Heine–Borel theorem starts in the 19th century, with the search for solid foundations of real analysis. Central to the theory was the concept of uniform continuity and the theorem stating that every continuous function on a closed interval is uniformly continuous. Peter Gustav Lejeune Dirichlet was the first to prove this and implicitly he used the existence of a finite subcover of a given open cover of a closed interval in his proof. He used this proof in his 1852 lectures, which were published only in 1904. Later Eduard Heine, Karl Weierstrass and Salvatore Pincherle used similar techniques. Émile Borel in 1895 was the first to state and prove a form of what is now called the Heine–Borel theorem. His formulation was restricted to countable covers. Pierre Cousin (1895), Lebesgue (1898) and Schoenflies (1900) generalized it to arbitrary covers.
 == Proof ==
@@ Line 13: / Line 13: @@
 '''If a set is compact, then it must be closed.'''
-Let ''S'' be a subset of '''R'''<sup>''n''</sup>.  Observe first the following: if ''a'' is a [[limit point]] of ''S'', then any finite collection ''C'' of open sets, such that each open set ''U'' ∈ ''C'' is disjoint from some [[Neighbourhood (mathematics)#Definition|neighborhood]] ''V''<sub>''U''</sub> of ''a'', fails to be a cover of ''S''.  Indeed, the intersection of the finite family of sets ''V''<sub>''U''</sub> is a neighborhood ''W'' of ''a'' in '''R'''<sup>''n''</sup>. Since ''a'' is a limit point of ''S'', ''W'' must contain a point ''x'' in ''S''. This ''x'' ∈ ''S'' is not covered by the family ''C'', because every ''U'' in ''C'' is disjoint from ''V''<sub>''U''</sub> and hence disjoint from ''W'', which contains ''x''.
+Let ''S'' be a subset of '''R'''<sup>''n''</sup>.  Observe first the following: if ''a'' is a limit point of ''S'', then any finite collection ''C'' of open sets, such that each open set ''U'' ∈ ''C'' is disjoint from some neighborhood ''V''<sub>''U''</sub> of ''a'', fails to be a cover of ''S''.  Indeed, the intersection of the finite family of sets ''V''<sub>''U''</sub> is a neighborhood ''W'' of ''a'' in '''R'''<sup>''n''</sup>. Since ''a'' is a limit point of ''S'', ''W'' must contain a point ''x'' in ''S''. This ''x'' ∈ ''S'' is not covered by the family ''C'', because every ''U'' in ''C'' is disjoint from ''V''<sub>''U''</sub> and hence disjoint from ''W'', which contains ''x''.
-If ''S'' is compact but not closed, then it has a limit point ''a'' not in ''S''.  Consider a collection {{nowrap|''C''&thinsp;′}} consisting of an open neighborhood ''N''(''x'') for each ''x'' ∈ ''S'', chosen small enough to not intersect some neighborhood ''V''<sub>''x''</sub> of ''a''.  Then {{nowrap|''C''&thinsp;′}} is an open cover of ''S'', but any finite subcollection of {{nowrap|''C''&thinsp;′}} has the form of ''C'' discussed previously, and thus cannot be an open subcover of ''S''.  This contradicts the compactness of ''S''.  Hence, every limit point of ''S'' is in ''S'', so ''S'' is closed.
+If ''S'' is compact but not closed, then it has a limit point ''a'' not in ''S''.  Consider a collection ''C''&thinsp;′ consisting of an open neighborhood ''N''(''x'') for each ''x'' ∈ ''S'', chosen small enough to not intersect some neighborhood ''V''<sub>''x''</sub> of ''a''.  Then ''C''&thinsp;′ is an open cover of ''S'', but any finite subcollection of ''C''&thinsp;′ has the form of ''C'' discussed previously, and thus cannot be an open subcover of ''S''.  This contradicts the compactness of ''S''.  Hence, every limit point of ''S'' is in ''S'', so ''S'' is closed.
-The proof above applies with almost no change to showing that any compact subset ''S'' of a [[Hausdorff space|Hausdorff]] topological space ''X'' is closed in ''X''.
+The proof above applies with almost no change to showing that any compact subset ''S'' of a Hausdorff topological space ''X'' is closed in ''X''.
 '''If a set is compact, then it is bounded.'''
@@ Line 56: / Line 56: @@
 == Heine–Borel property ==
-The Heine–Borel theorem does not hold as stated for general [[Metric space|metric]] and [[topological vector space]]s, and this gives rise to the necessity to consider special classes of spaces where this proposition is true. They are called the '''spaces with the Heine–Borel property'''.
+The Heine–Borel theorem does not hold as stated for general metric and topological vector spaces, and this gives rise to the necessity to consider special classes of spaces where this proposition is true. They are called the '''spaces with the Heine–Borel property'''.
 ===In the theory of metric spaces===
-A [[metric space]] <math>(X,d)</math> is said to have the '''Heine–Borel property''' if each closed bounded<ref>A set <math>B</math> in a metric space <math>(X,d)</math> is said to be ''bounded'' if it is contained in a ball of a finite radius, i.e. there exists <math>x\in X</math> and <math>r>0</math> such that <math>B\subseteq\{x\in X: \ d(x,a)\le r\}</math>.</ref> set in <math>X</math> is compact.
+A metric space <math>(X,d)</math> is said to have the '''Heine–Borel property''' if each closed bounded set in <math>X</math> is compact.
-Many metric spaces fail to have the Heine–Borel property, such as the metric space of [[rational number]]s (or indeed any incomplete metric space). Complete metric spaces may also fail to have the property; for instance, no infinite-dimensional [[Banach space]]s have the Heine–Borel property (as metric spaces). Even more trivially, if the real line is not endowed with the usual metric, it may fail to have the Heine–Borel property.
+Many metric spaces fail to have the Heine–Borel property, such as the metric space of rational numbers (or indeed any incomplete metric space). Complete metric spaces may also fail to have the property; for instance, no infinite-dimensional Banach spaces have the Heine–Borel property (as metric spaces). Even more trivially, if the real line is not endowed with the usual metric, it may fail to have the Heine–Borel property.
-A metric space <math>(X,d)</math> has a Heine–Borel metric which is Cauchy locally identical to <math>d</math> if and only if it is [[complete space|complete]], [[sigma-compact|<math>\sigma</math>-compact]], and [[locally compact space|locally compact]].{{sfn|Williamson|Janos|1987}}
+A metric space <math>(X,d)</math> has a Heine–Borel metric which is Cauchy locally identical to <math>d</math> if and only if it is complete, <math>\sigma</math>-compact, and locally compact.
 ===In the theory of topological vector spaces===
-A [[topological vector space]] <math>X</math> is said to have the '''Heine–Borel property'''{{sfn|Kirillov|Gvishiani|1982|loc=Theorem 28}} (R.E. Edwards uses the term ''boundedly compact space''{{sfn|Edwards|1965|loc=8.4.7}}) if each closed bounded<ref>A set <math>B</math> in a topological vector space <math>X</math> is said to be ''bounded'' if for each neighborhood of zero <math>U</math> in <math>X</math> there exists a scalar <math>\lambda</math> such that <math>B\subseteq\lambda\cdot U</math>.</ref> set in <math>X</math> is compact.<ref>In the case when the topology of a topological vector space <math>X</math> is generated by some metric <math>d</math> this definition is not equivalent to the definition of the Heine–Borel property of <math>X</math> as a metric space, since the notion of bounded set in <math>X</math> as a metric space is different from the notion of bounded set in <math>X</math> as a topological vector space. For instance, the space <math>{\mathcal C}^\infty[0,1]</math> of smooth functions on the interval <math>[0,1]</math> with the metric <math>d(x,y)=\sum_{k=0}^\infty\frac{1}{2^k}\cdot\frac{\max_{t\in[0,1]}|x^{(k)}(t)-y^{(k)}(t)|}{1+\max_{t\in[0,1]}|x^{(k)}(t)-y^{(k)}(t)|}</math> (here <math>x^{(k)}</math> is the <math>k</math>-th derivative of the function <math>x\in {\mathcal C}^\infty[0,1]</math>) has the Heine–Borel property as a topological vector space but not as a metric space.</ref> No infinite-dimensional [[Banach space]]s have the Heine–Borel property (as topological vector spaces). But some infinite-dimensional [[Fréchet space]]s do have, for instance, the space <math>C^\infty(\Omega)</math> of smooth functions on an open set <math>\Omega\subset\mathbb{R}^n</math>{{sfn|Edwards|1965|loc=8.4.7}} and the space <math>H(\Omega)</math> of holomorphic functions on an open set <math>\Omega\subset\mathbb{C}^n</math>.{{sfn|Edwards|1965|loc=8.4.7}}  More generally, any quasi-complete [[nuclear space]] has the Heine–Borel property. All [[Montel space]]s have the Heine–Borel property as well.
+A topological vector space <math>X</math> is said to have the '''Heine–Borel property''' (R.E. Edwards uses the term ''boundedly compact space'') if each closed bounded set in <math>X</math> is compact. No infinite-dimensional Banach spaces have the Heine–Borel property (as topological vector spaces). But some infinite-dimensional Fréchet spaces do have, for instance, the space <math>C^\infty(\Omega)</math> of smooth functions on an open set <math>\Omega\subset\mathbb{R}^n</math> and the space <math>H(\Omega)</math> of holomorphic functions on an open set <math>\Omega\subset\mathbb{C}^n</math>. More generally, any quasi-complete nuclear space has the Heine–Borel property. All Montel spaces have the Heine–Borel property as well.
 ==Licensing==
 Content obtained and/or adapted from:
 * [https://en.wikipedia.org/wiki/Heine%E2%80%93Borel_theorem Heine-Borel theorem, Wikipedia] under a CC BY-SA license

Heine-Borel Theorem - Revision history

Lila at 19:36, 27 October 2021

Lila at 19:35, 27 October 2021

Lila: Created page with "In real analysis the '''Heine–Borel theorem''', named after Eduard Heine and Émile Borel, states: For a subset ''S'' of Euclidean space '''R'''''n..."

Lila: Created page with "In real analysis the '''Heine–Borel theorem''', named after Eduard Heine and Émile Borel, states: For a subset ''S'' of Euclidean space '''R'''^''n..."