Ascoli's formula

The formula of Ascoli ( English formula of Ascoli ) is a mathematical formula that in one of the Italian mathematician Guido Ascoli presented in 1932, work back and in the transition area between the areas functional analysis and geometry is located. It gives a description of the distance between a point in space and a given affine hyperplane in a real normalized space .

Representation of the formula

The formula can be specified as follows:

Given a normalized - vector space and its dual space of real-valued continuous linear functional , with both the standard of and the operator norm of having to be called. ${\displaystyle \mathbb {R} }$ ${\displaystyle X}$ ${\displaystyle X^{*}}$ ${\displaystyle X}$ ${\displaystyle X^{*}}$ ${\displaystyle {\|\cdot \|}}$

An affine hyperplane is also given , with a real number and a functional . ${\displaystyle H\subset X}$ ${\displaystyle H=\{x\in X\mid u^{*}(x)=\alpha \}}$ ${\displaystyle \alpha }$ ${\displaystyle u^{*}\in {X^{*}\setminus \{0\}}}$

Then the distance between it and the hyperplane is calculated for any point in space using the formula ${\displaystyle a\in X}$ ${\displaystyle d(a,H)=\inf _{h\in H}{\|h-a\|}}$

${\displaystyle d(a,H)={\frac {|u^{*}(a)-\alpha |}{\|u^{*}\|}}}$.

Direct evidence

Subsequent to the presentation in Ivan Singer's monograph, direct evidence can be given in the following way:

First is for anything ${\displaystyle h\in H}$

${\displaystyle u^{*}(h)=\alpha }$

and thus - due to the properties of the operator norm! -

${\displaystyle |u^{*}(a)-\alpha |=|u^{*}(a-h)|\leq \|u^{*}\|\cdot \|a-h\|}$

and therefore

${\displaystyle \|h-a\|\geq {\frac {|u^{*}(a)-\alpha |}{\|u^{*}\|}}}$.

So the inequality holds

${\displaystyle d(a,H)\geq {\frac {|u^{*}(a)-\alpha |}{\|u^{*}\|}}}$.

To prove the reverse inequality, one takes into account that - again due to the properties of the operator norm! - the relationship

${\displaystyle 0<{\|u^{*}\|}=\sup _{x\in X\;,\;\|x\|=1}{|u^{*}(x)|}}$

exists, and thus for every real number with is always a with and . ${\displaystyle r}$ ${\displaystyle 0<r<\|u^{*}\|}$ ${\displaystyle x_{r}\in X}$ ${\displaystyle \|x_{r}\|=1}$ ${\displaystyle |u^{*}(x_{r})|>\|u^{*}\|-r}$

For this is

${\displaystyle h_{r}:=a-{\frac {u^{*}(a)-\alpha }{u^{*}(x_{r})}}\cdot x_{r}}$

set. Apparently and is there ${\displaystyle h_{r}\in H}$

${\displaystyle d(a,H)\leq \|h_{r}-a\|=|{\frac {u^{*}(a)-\alpha }{u^{*}(x_{r})}}|\cdot 1={\frac {|u^{*}(a)-\alpha |}{|u^{*}(x_{r})|}}<{\frac {|u^{*}(a)-\alpha |}{\|u^{*}\|-r}}}$.

By crossing the border you finally win ${\displaystyle r\to 0}$

${\displaystyle d(a,H)\leq {\frac {|u^{*}(a)-\alpha |}{\|u^{*}\|}}}$.

The formula proves it.

background

The Ascoli'sche formula can also be obtained from the so-called duality theorem of linear approximation theory ( English duality theorem of linear approximation theory ) win the following states:

Be , and given above. ${\displaystyle X}$ ${\displaystyle X^{*}}$ ${\displaystyle \|\cdot \|}$

Let there be a subspace and a point in space . ${\displaystyle V\subseteq X}$ ${\displaystyle a\in X}$

Let the orthogonal complement of in . ${\displaystyle V^{\perp }=\{x^{*}\in X^{*}\mid \forall v\in V\colon x^{*}(v)=0\}}$ ${\displaystyle V}$ ${\displaystyle X^{*}}$

Then the formula applies to the distance between point in space and subspace ${\displaystyle d(a,V)=\inf _{v\in V}{\|v-a\|}}$

${\displaystyle d(a,V)=\max _{x^{*}\in V^{\perp }{\text{ und }}\|x^{*}\|\leq 1}{x^{*}(a)}}$.

Explanations and Notes

• The question on which Ascoli's formula is based is closely related to the problem posed in analytical geometry in connection with Hessian normal form , how to calculate the Euclidean distance of a point from a straight line in or from a plane in . ${\displaystyle {\mathbb {R} }^{2}}$ ${\displaystyle {\mathbb {R} }^{3}}$
• If above is for a point in space , then the distance treated in Ascoli's formula is also calculated according to the formula . ${\displaystyle u^{*}(h_{0})=\alpha }$ ${\displaystyle h_{0}\in H}$ ${\displaystyle d(a,H)={\frac {|u^{*}(a-h_{0})|}{\|u^{*}\|}}}$
• According to a general theorem of the mathematician Werner Fenchel , the maximum occurring in the above duality theorem of the linear approximation theory always exists.

Sample calculation

In the vector space , the distance is to be calculated for the plane and the point in space according to different standards ; namely for the Euclidean norm , the sum norm and the maximum norm . The following distances are obtained here: ${\displaystyle {\mathbb {R} }^{3}=\{(x_{1},x_{2},x_{3})\mid x_{1},x_{2},x_{3}\in \mathbb {R} \}}$ ${\displaystyle H=\{(x_{1},x_{2},x_{3})\in {\mathbb {R} }^{3}\mid x_{1}+3x_{2}+x_{3}=1\}}$ ${\displaystyle a=(1,2,3)}$ ${\displaystyle d(a,H)}$ ${\displaystyle \|\cdot \|_{2}}$ ${\displaystyle \|\cdot \|_{1}}$ ${\displaystyle \|\cdot \|_{\max }}$

(a) For : ${\displaystyle \|\cdot \|_{2}}$

${\displaystyle d(a,H)={\frac {|1\cdot 1+3\cdot 2+1\cdot 3-1|}{\sqrt {1+3^{2}+1}}}={\frac {9}{\sqrt {11}}}}$

(b) For : ${\displaystyle \|\cdot \|_{1}}$

${\displaystyle d(a,H)={\frac {9}{\max\{1,3,1\}}}={\frac {9}{3}}}$

(c) For : ${\displaystyle \|\cdot \|_{\max }}$

${\displaystyle d(a,H)={\frac {9}{1+3+1}}={\frac {9}{5}}}$

literature

• Guido Ascoli: Sugli spazi lineari metrici e le loro varietà lineari . In: Annali di Matematica Pura ed Applicata . tape 10 , 1932, p. 33-81, 203-232 ( MR1553181 ).
• Gerd Fischer : Linear Algebra . An introduction for first-year students. 16th, revised and expanded edition. Vieweg + Teubner , Wiesbaden 2008, ISBN 978-3-8348-0428-0 , p. 280-281 .
• Peter Kosmol : Optimization and Approximation (=  De Gruyter Studies ). 2nd Edition. Walter de Gruyter & Co. , Berlin 2010, ISBN 978-3-11-021814-5 ( MR2599674 ).
• Peter Kosmol, Dieter Müller-Wichards : Optimization in Function Spaces . With stability considerations in Orlicz spaces (=  De Gruyter Series in Nonlinear Analysis and Applications . Volume 13 ). Walter de Gruyter & Co., Berlin 2011, ISBN 978-3-11-025020-6 ( MR2760903 ).
• Ivan Singer: Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces . Translation of the original Romanian version "Cea mai bună aproximare în spații vectoriale normate prin elemente din subspații vectoriale". Translated by Radu Georgescu (=  The basic teachings of the mathematical sciences in individual representations with special consideration of the areas of application . Volume 171 ). Springer Verlag, Berlin, Heidelberg, New York 1970 ( MR0270044 ).

See also

• Hessian normal form

Individual evidence

1. ↑ Peter Kosmol: Optimization and Approximation. 2010, pp. 399-400
2. ↑ ^{a b c} Peter Kosmol, Dieter Müller-Wichards: Optimization in Function Spaces. 2011, SS 108
3. ↑ ^{a b} Ivan Singer: Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces. 1970, p. 24
4. ↑ ^{a b} Kosmol, op.cit., P. 400
5. ↑ is the amount function . ${\displaystyle |\cdot |}$
6. ↑ Kosmol, op.cit., P. 399
7. ↑ Kosmol, op.cit., P. 385, p. 399