Editing Heine–Borel theorem (section)

===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.