Osgood's theorem (functional analysis)

The set of Osgood is a mathematical theorem that in the transition area between functional analysis and topology settled and after the mathematician William Fogg Osgood is named. It is closely related to, and even a direct consequence of, Baire's category theorem . The consequence of Osgood's theorem is the principle of uniform limitation , one of the classic results of functional analysis.

Formulation of the sentence

A topological space is given and in it a subset of the second Bairean category . ${\displaystyle \left(X,{\mathcal {O}}\right)}$ ${\displaystyle W\subseteq X}$

Let a family of real-valued functions continuous below be given ${\displaystyle {\mathcal {F}}=(f_{i})_{i\in I}}$

${\displaystyle f_{i}\colon \left(X,{\mathcal {O}}\right)\to \mathbb {R} \;\;(i\in I)}$.

For this it is assumed that the family is limited to pointwise evenly upwards : ${\displaystyle {\mathcal {F}}}$ ${\displaystyle W}$

${\displaystyle \sup _{i\in I}{f_{i}(w)}<\infty \;\;(w\in W)}$

Then:

There is a non-empty open subset in such a way that the family of restricted functions is evenly bounded upwards , i.e. the condition ${\displaystyle U\in {\mathcal {O}}}$ ${\displaystyle {\mathcal {F}}{\upharpoonright U}={({f_{i}}{\upharpoonright }U)}_{i\in I}}$ ${\displaystyle U}$

${\displaystyle \sup _{i\in I,u\in U}{f_{i}(u)}<\infty }$

enough.

Evidence sketch

Under the conditions mentioned, the supremum function , defined by the assignment rule , is even below continuous below ${\displaystyle F=\sup {\mathcal {F}}\colon \left(X,{\mathcal {O}}\right)\to \mathbb {R} }$ ${\displaystyle x\mapsto \sup _{i\in I}f_{i}(x)}$

Consequently, it is sufficient to carry out the proof only for the case of a single real function which is continuous below . In addition, one can assume from the outset . ${\displaystyle f}$ ${\displaystyle X=W}$

Now form the subset for each . ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle W_{n}=\{x\in X:f(x)\leq n\}}$

This form because of the mentioned half continuity condition one out of sheer closed sets existing coverage of . Since, according to the assumption, is of the second Bairean category , one of these closed subsets, for example , necessarily has a non-empty interior . Now you bet and win the open subset you are looking for. ${\displaystyle {(W_{n})}_{n\in \mathbb {N} }}$ ${\displaystyle W}$ ${\displaystyle X=W}$ ${\displaystyle W_{N}}$ ${\displaystyle {U=W_{N}}^{\circ }}$

Inferences

With Osgood's theorem one arrives directly at the principle of uniform restriction , according to which:

A family of continuous operators bounded pointwise upwards from a Banach space into a normalized space is always bounded uniformly upwards with respect to the operator norm .

First of all, Osgood's theorem is particularly applicable to the case that a Bairean space is and that all are continuous . According to the Bairean category theorem, it is certainly also valid if its topological structure is generated by a complete metric . In this case, the statement can be tightened and one obtains the following special version of Osgood's theorem : ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle f_{i}}$   ${\displaystyle (i\in I)}$

If there is a point-wise upwardly bounded family of continuous real-valued functions on a complete metric space , then there exists a closed full sphere with . ${\displaystyle (f_{i})_{i\in I}}$ ${\displaystyle (X,d)}$ ${\displaystyle {\overline {B_{r}}}\subseteq X}$ ${\displaystyle \sup _{i\in I,x\in {\overline {B_{r}}}}{f_{i}(x)}<\infty }$

Starting from this special version, one can tighten it further by taking into account the special uniform structure of standardized spaces , according to which all closed full spheres emerge from the closed unit sphere through centric extension and parallel displacement. If one also takes into account that the concatenation of a continuous operator with a norm - if possible - always results in a real-valued function that is below continuous , then one has the principle of uniform boundedness .

Related result

Closely related to the above theorems is the following lemma by Gelfand :

Be a normed space and is one of semi-continuous bottom seminorm on . If this seminorm is bounded pointwise upwards to a subset of the second Bairean category , then there is a real constant with: ${\displaystyle (X,\|\cdot \|)}$ ${\displaystyle p:(X,\|\cdot \|)\to \mathbb {R} }$ ${\displaystyle (X,\|\cdot \|)}$ ${\displaystyle M>0}$

${\displaystyle p(x)\leq M\|x\|}$   ${\displaystyle (x\in X)}$.

The lemma can be derived from Osgood's theorem with the same considerations as above . For its part (and in the same way as above) it leads directly to the principle of uniform limitation .

Further conclusions from the principle of uniform limitation

• Every weakly bounded subset - thus every weakly convergent sequence - in a normalized vector space is bounded .
• If a family of continuous linear operators of a Banach space is in a normalized space and is weakly bounded in for each , then it is bounded uniformly upwards with respect to the operator norm. ${\displaystyle {\mathcal {F}}}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle {\mathcal {F}}(x)=\{f(x):f\in {\mathcal {F}}\}}$ ${\displaystyle x\in X}$ ${\displaystyle Y}$ ${\displaystyle {\mathcal {F}}}$
• If a sequence of continuous linear operators converges from a Banach space into a normalized space point by point against a limit function , then is also a continuous linear operator and applies ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle f}$ ${\displaystyle f}$

${\displaystyle \|f\|\leq \liminf _{n\rightarrow \infty }\|f_{n}\|}$.

literature

Monographs

• Izrail M. Gelfand : Collected Papers. Volume I. Edited by SG Gindikin - VW Guillemin - AA Kirillov - B. Kostant - S. Sternberg . Springer Verlag, Berlin a. a. 1987, ISBN 3-540-13619-3 .
• Harro Heuser : Functional Analysis. Theory and application (=  mathematical guidelines ). 4th revised edition. Teubner Verlag, Wiesbaden 2006, ISBN 3-8351-0026-2 ( MR2380292 ).
• LW Kantorowitsch , GP Akilow : Functional analysis in standardized spaces. Edited in German by Prof. Dr. rer. nat. habil. P. Heinz Müller, Technical University of Dresden. Translated from Russian by Heinz Langer, Dresden, and Rolf Kühne, Dresden . Verlag Harri Deutsch, Thun / Frankfurt am Main 1978, ISBN 3-87144-327-1 ( MR0458199 ).
• Ronald Larsen: Functional Analysis. An Introduction (=  Pure and Applied Mathematics . Volume 15 ). Marcel Dekker, New York 1973, ISBN 0-8247-6042-5 ( MR0461069 ).
• Horst Schubert : Topology . 4th edition. BG Teubner Verlag, Stuttgart 1975, ISBN 3-519-12200-6 ( MR0423277 ).

Individual evidence

1. ↑ ^{a b} H. Heuser: functional analysis. Theory and application . 2006, p. 246 ff .
2. ^ ^{A b} R. Larsen: Functional Analysis. An Introduction . 1973, p. 146 ff .
3. ^ H. Schubert: Topology . 1975, p. 132 ff .
4. ^ H. Schubert: Topology . 1975, p. 134 .
5. ↑ It is even a seminar norm.
6. ^ IM Gelfand: Collected Papers . 1987, p. 205 ff .
7. ↑ ^{a b} L. W. Kantorowitsch, GP Akilow: functional analysis in standardized spaces . 1978, p. 107 ff .
8. ↑ Kantorowitsch and Akilow call such a seminorm a downward semi-continuous convex functional . ${\displaystyle p}$
9. ^ H. Heuser: functional analysis. Theory and application . 2006, p. 326 .
10. ↑ ^{a b} F. Hirzebruch, W. Scharlau: Title? Year ?, pp. 37–38
11. ↑ This follows with the principle of uniform restriction due to the fact that every normalized space can be embedded linearly-isometrically in its dual space .
12. ^ H. Heuser: functional analysis. Theory and application . 2006, p. 248 .