Lemma of Zabreiko

The lemma Zabreiko , named after Petr Petrovich Zabreiko ( Russian Петр Петрович Забрейко ), is a statement from the mathematical branch of functional analysis . It dates from 1969 and is a statement of continuity about certain subadditive functionals on Banach spaces .

Formulation of the lemma

Let it be a Banach space and a functional with the following properties: ${\displaystyle X}$ ${\displaystyle p:X\rightarrow [0,\infty )}$

• ${\displaystyle \lim _{t\searrow 0}p(tx)=0}$   for all   , ${\displaystyle x\in X}$
• ${\displaystyle p(\sum _{n=0}^{\infty }x_{n})\leq \sum _{n=0}^{\infty }p(x_{n})}$   for every convergent series   in . ${\displaystyle \sum _{n=0}^{\infty }x_{n}}$ ${\displaystyle X}$

Then is steady. ${\displaystyle p}$

Remarks

Subadditivity follows in particular from the first property and then from the second ${\displaystyle p(0)=0}$

${\displaystyle p(x+y)\leq p(x)+p(y)}$for all , ${\displaystyle x,y\in X}$

by looking at the series with , and for all . ${\displaystyle \sum _{n=0}^{\infty }x_{n}}$ ${\displaystyle x_{0}=x}$ ${\displaystyle x_{1}=y}$ ${\displaystyle x_{n}=0}$ ${\displaystyle n>1}$

The proof uses the completeness of the Banach space in the form of Baire's theorem . Zabreiko's lemma cannot be proven for incomplete normed spaces .

The meaning of the lemma results from the fact that the so-called three principles of functional analysis, i.e. the theorem of uniform boundedness , the theorem of the open mapping and the theorem of the closed graph , all of which are classically based on Baire's theorem, are easy Zabreiko's lemma can be traced back without having to bring Baire's theorem back into play. This structure of the functional analysis is detailed in the textbooks by VI Istrățescu and RE Megginson.

application

We show here an example of how the theorem of the closed graph can be derived from Zabreiko's lemma:

Let be a linear operator between Banach spaces and let its graph be closed . We want to show the continuity of : ${\displaystyle T:X\rightarrow Y}$ ${\displaystyle T}$

Look at the functional . Obviously it is enough to show the continuity of and we want to do that with the Zabreiko lemma. apparently fulfills the first condition from Zabreiko's lemma. To prove the second condition, let convergent in . It is to be shown what is clear when the right side is infinite. If the right hand side is finite then there is absolute convergence and because of the completeness of there is a with . Then and is such that because of the presupposed closure of the graph lies in the graph of and that means . So is ${\displaystyle p:X\rightarrow \mathbb {R} ,\,p(x):=\|Tx\|}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle \sum _{n=0}^{\infty }x_{n}=x}$ ${\displaystyle X}$ ${\displaystyle \|T(\sum _{n=0}^{\infty }x_{n})\|\leq \sum _{n=0}^{\infty }\|T(x_{n})\|}$ ${\displaystyle Y}$ ${\displaystyle y\in Y}$ ${\displaystyle y=\sum _{n=0}^{\infty }T(x_{n})}$ ${\displaystyle \sum _{n=0}^{m}x_{n}\,{\xrightarrow {m}}\,x}$ ${\displaystyle T(\sum _{n=0}^{m}x_{n})=\sum _{n=0}^{m}T(x_{n})\,{\xrightarrow {m}}\,y}$ ${\displaystyle (x,y)}$ ${\displaystyle T}$ ${\displaystyle Tx=y}$

${\displaystyle p(\sum _{n=0}^{\infty }x_{n})=\|T(x)\|=\|y\|=\|\sum _{n=0}^{\infty }T(x_{n})\|\leq \sum _{n=0}^{\infty }\|T(x_{n})\|=\sum _{n=0}^{\infty }p(x_{n})}$.

With this the lemma of Zabreiko can be applied, because it is also a Banach space, and the continuity of follows . That ends the derivation of the theorem from the closed graph. ${\displaystyle X}$ ${\displaystyle p}$

Individual evidence

1. ↑ П. П. Забрейко: Об одной теореме для полуаддитивных функционалов , Функциональный анализ и его приложения, (1969), Volume 3, Number 1 (1969), pages 86-88 (German: PP Sabreiko: About one set for halbaddiditve Functional )
2. ↑ PP Zabreiko: A theorem for semiadditive functionals , Functional analysis and its applications (1969), Volume 3, Number 1, Pages 70-72
3. ^ Vasile I. Istrățescu: Strict convexity and complex strict convexity , Lecture Notes in Pure and Applied Mathematics, Volume 89, Marcel Dekker (1984), ISBN 0-8247-1796-1
4. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , 1.6.3: Zabreiko's Lemma , (here only for homogeneous functionals)
5. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , proof of 1.6.11: The Closed Graph Theorem