# Normed Koethe Spaces as Intermediate Spaces of L(1) and L

## Transcript Of Normed Koethe Spaces as Intermediate Spaces of L(1) and L

Louisiana State University

LSU Digital Commons

LSU Historical Dissertations and Theses

Graduate School

1972

Normed Koethe Spaces as Intermediate Spaces of L(1) and L(,infinity).

Stuart Edward Mills Louisiana State University and Agricultural & Mechanical College

Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_disstheses

Recommended Citation Mills, Stuart Edward, "Normed Koethe Spaces as Intermediate Spaces of L(1) and L(,infinity)." (1972). LSU Historical Dissertations and Theses. 2299. https://digitalcommons.lsu.edu/gradschool_disstheses/2299

This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Historical Dissertations and Theses by an authorized administrator of LSU Digital Commons. For more information, please contact [email protected]

INFORMATION TO USERS

This dissertation was produced from a m icrofilm copy of the original docum ent. W hile the most advanced technological means to photograph and reproduce this d o c u m e n t have been used, th e q u a lity is heavily d ep e n d e n t upon th e q u a lity o f the original subm itted.

T h e fo llo w in g e x p la n a tio n o f techniques is p ro vid ed to help you understand markings or patterns which may appear on this reproduction.

1. T h e sign or " ta rg e t" fo r pages a p p a re n tly lacking fro m the d o cu m e n t p h o to g ra p h e d is "M issing Page(s)". If it was possible to o b ta in the missing page(s) or section, they are spliced in to the film along w ith adjacen t pages. T his m ay have necessitated c u ttin g th ru an image and d u p lic a tin g adjacen t pages to insure y o u c o m p le te c o n tin u ity .

2. W hen an image on th e film is o b lite ra te d w ith a large round black m a rk, it is an in d ic atio n th a t th e p h o to g ra p h e r suspected th a t the copy m ay have m oved during exposure and thus cause a blurred image. Y o u w ill fin d a good image o f th e page in the adjacent fram e.

3. When a map, drawing or chart, etc., was part o f the m aterial being p h o to g r a p h e d th e p h o to g ra p h e r fo llo w e d a d e fin ite m e th o d in "s e c tio n in g " th e m aterial. It is c u sto m ary to begin p h o to in g at th e upper le ft hand corner o f a large sheet and to con tin u e photo in g fro m 'e ft to right in equal sections w ith a small overlap. If necessary, sectioning is c o n tin u e d again — beginning b e lo w th e firs t ro w and continuing on until com plete.

4. T h e m a jo rity o f users in d icate th a t th e te x tu a l c o n te n t is o f greatest value, however, a som ew hat higher q u ality reproduction could be made from "ph o tographs" if essential to the understanding o f the dissertation. Silver prints of "photographs" may be ordered at ad d itio n al charge by w ritin g the O rder D ep artm en t, giving the catalog num ber, title, a u th o r and specific pages you wish reproduced.

University Microfilms

300 North Zeeb Road Ann Arbor. Michigan 48106 A Xerox Education Company

73-2972 MILLS, Stuart Edward, 19^6-

NORMED KtfTHE SPACES AS INTERMEDIATE SPACES OF Li AND L qo . The Louisiana State University and Agricultural and Mechanical College, Ph.D., 1972 Mathematics University Microfilms, A XEROX Company , Ann Arbor, Michigan

T H IS D IS S E R T A T IO N HAS BEEN M IC R O F IL M E D E X A C T L Y AS R E C EIVED .

Normed Kothe Spaces as Intermediate

Spaces of L, and L

r

1

a

A Dissertation

Submitted to the Graduate Faculty of the Louisiana State University and

Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy

in

The Department of Mathematics

B.S., M.S.,

by Stuart Edward Mills Louisiana State University, Louisiana State University,

August, 1972

1968 1970

PLEASE NOTE:

Some pages may have indistinct print. Filmed as received.

University Microfilms, A Xerox Education Company

ACKNOWLEDGEMENT The author wishes to express his sincere gratitude to Professor James R. Dorroh for his guidance and advice through out the past three years. Also the author wishes to thank Professor Dorroh for the assistance which he gave so freely throughout the preparation of this work. Further, the author wishes to express his appreciation to his wife, Pat, for her encouragement, understanding, and sacrifice through these years of schooling. In addition, the author wants to thank his wife for typing this manuscript.

ii

TABLE OF CONTENTS

CHAPTER I; INTRODUCTION

1. Statement of the Problem

2. Preliminaries

CHAPTER II: THE L1 +

NORM

CHAPTER III: ORLICZ SPACES AS INTERMEDIATE SPACES

1. Basic Properties of Orlicz Spaces

2. L^ O L^, L^ + L^, and Orlicz Space

3. Monotonic Rearrangement

CHAPTER IV: REARRANGEMENT INVARIANT KOTHE SPACES

1. Normed Spaces

2. Universally Rearrangement Invariant Function

Norms

3. Universal and Universally Rearrangement

Invariant Kothe Spaces

BIBLIOGRAPHY

VITA

Page 1 1 3 9

19 19 21 33 38 38

59

67 72 74

iii

ABSTRACT

Let X^ and

be two Banach spaces contained in a linear

Hausdorff space Y such that the identity mapping of X^(i=l,2)

in Y is continuous. Denote the elements of X^ by f^ and

their norms by IlfII^ . The spaces X^ O

and X^ +

are

Banach spaces under the norms ^ ^ Xj_OX2 = m3X ^ ^ 1 * ^ ^ 2 ^

||f|L ,Y - inf 1 2 f-f1+f2

(||f,||, + ||folL) • A Banach space X C Y

11

1 l

satisfyingX j C X C X j + X2

and ||f||x +x^ s:||f||x *

is called an intermediatespace of X^ and X^ *

an£* X2

Let (A,E,p) be a totally o-finite measure space and let

M(A) be the set of all complex-valued y-measurable functions on

A . Then M(A) is a linear Hausdorff space under convergence in

measure on sets of finite measure. This dissertation is concerned

with determining whether certain classes of norraed Kbthe spaces

(Banach function spaces) are intermediate spaces of L^ = L^(y)

and

L * L (p) .

00

00

It is *proven that

LX. D

00LandL, +X L

0a0re

associate Orlicz spaces and that for every non-trivial Young's

function T there is an equivalent Young's function T' such

that the Orlicz space L ^ , is an intermediate space of L^ and

L^ . The concept of universal function norm is introduced and

it is proven that p is induced by a universal function norm if

and only if

p isa universally rearrangement invariant function iv

norm if and only if p(f) has a representation in terms of f ,

the non-increasing rearrangement of f . The notion of a universal

Kdthe space is presented and it is proven that a KSthe space is

universal if and only if it is universally rearrangement invariant.

It is proven that if A is a universal Kothe space then

L,1 fl L00 C a C L .1 + L00 . Furthermore, if A is normed, in partic-

ular A =

, then there is an equivalent universally rearrange

ment invariant norm p, for which L

is an intermediate space

1

pi

of L, and L

1

00

v

CHAPTER I: INTRODUCTION

1.

Statement of the Problem. Let X^ and X 2 be two Banach

spaces contained in a linear Hausdroff space Y such that the in

jection of X^ (i * 1,2) into Yis continuous. Denote the norm of

by || ||^. The space X^ H X2 is the set of all elements which

are in both X^ and X 2 , and the space X^ + X 2 is the set of all

f e Y of the form f ■ f^ + f2 with f^ e X^ and

e X2 . It is

known that thespaces X^ O X2 and X^ + X2

are Banach spaces under

the norms ||f|fx p x “ maxt ||f • ||f|(2 > and ||fffx + x =

inf{||f1||1 + ||f2II2 : f - £x + f2 , £t e 3.2.1]).

(see [1, p. 165, Prop.

Definition 1.1: A Banach space X C Y satisfying

xxn x2c x C x 1 + x2

and

+ X2 s ««», s »Xln X2

is called an intermediate space of X^ and X2 . Much work has been done on intermediate spaces and the related

topic of interpolation theory. (See [1], [2], [16].) In particular, 1

LSU Digital Commons

LSU Historical Dissertations and Theses

Graduate School

1972

Normed Koethe Spaces as Intermediate Spaces of L(1) and L(,infinity).

Stuart Edward Mills Louisiana State University and Agricultural & Mechanical College

Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_disstheses

Recommended Citation Mills, Stuart Edward, "Normed Koethe Spaces as Intermediate Spaces of L(1) and L(,infinity)." (1972). LSU Historical Dissertations and Theses. 2299. https://digitalcommons.lsu.edu/gradschool_disstheses/2299

This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Historical Dissertations and Theses by an authorized administrator of LSU Digital Commons. For more information, please contact [email protected]

INFORMATION TO USERS

This dissertation was produced from a m icrofilm copy of the original docum ent. W hile the most advanced technological means to photograph and reproduce this d o c u m e n t have been used, th e q u a lity is heavily d ep e n d e n t upon th e q u a lity o f the original subm itted.

T h e fo llo w in g e x p la n a tio n o f techniques is p ro vid ed to help you understand markings or patterns which may appear on this reproduction.

1. T h e sign or " ta rg e t" fo r pages a p p a re n tly lacking fro m the d o cu m e n t p h o to g ra p h e d is "M issing Page(s)". If it was possible to o b ta in the missing page(s) or section, they are spliced in to the film along w ith adjacen t pages. T his m ay have necessitated c u ttin g th ru an image and d u p lic a tin g adjacen t pages to insure y o u c o m p le te c o n tin u ity .

2. W hen an image on th e film is o b lite ra te d w ith a large round black m a rk, it is an in d ic atio n th a t th e p h o to g ra p h e r suspected th a t the copy m ay have m oved during exposure and thus cause a blurred image. Y o u w ill fin d a good image o f th e page in the adjacent fram e.

3. When a map, drawing or chart, etc., was part o f the m aterial being p h o to g r a p h e d th e p h o to g ra p h e r fo llo w e d a d e fin ite m e th o d in "s e c tio n in g " th e m aterial. It is c u sto m ary to begin p h o to in g at th e upper le ft hand corner o f a large sheet and to con tin u e photo in g fro m 'e ft to right in equal sections w ith a small overlap. If necessary, sectioning is c o n tin u e d again — beginning b e lo w th e firs t ro w and continuing on until com plete.

4. T h e m a jo rity o f users in d icate th a t th e te x tu a l c o n te n t is o f greatest value, however, a som ew hat higher q u ality reproduction could be made from "ph o tographs" if essential to the understanding o f the dissertation. Silver prints of "photographs" may be ordered at ad d itio n al charge by w ritin g the O rder D ep artm en t, giving the catalog num ber, title, a u th o r and specific pages you wish reproduced.

University Microfilms

300 North Zeeb Road Ann Arbor. Michigan 48106 A Xerox Education Company

73-2972 MILLS, Stuart Edward, 19^6-

NORMED KtfTHE SPACES AS INTERMEDIATE SPACES OF Li AND L qo . The Louisiana State University and Agricultural and Mechanical College, Ph.D., 1972 Mathematics University Microfilms, A XEROX Company , Ann Arbor, Michigan

T H IS D IS S E R T A T IO N HAS BEEN M IC R O F IL M E D E X A C T L Y AS R E C EIVED .

Normed Kothe Spaces as Intermediate

Spaces of L, and L

r

1

a

A Dissertation

Submitted to the Graduate Faculty of the Louisiana State University and

Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy

in

The Department of Mathematics

B.S., M.S.,

by Stuart Edward Mills Louisiana State University, Louisiana State University,

August, 1972

1968 1970

PLEASE NOTE:

Some pages may have indistinct print. Filmed as received.

University Microfilms, A Xerox Education Company

ACKNOWLEDGEMENT The author wishes to express his sincere gratitude to Professor James R. Dorroh for his guidance and advice through out the past three years. Also the author wishes to thank Professor Dorroh for the assistance which he gave so freely throughout the preparation of this work. Further, the author wishes to express his appreciation to his wife, Pat, for her encouragement, understanding, and sacrifice through these years of schooling. In addition, the author wants to thank his wife for typing this manuscript.

ii

TABLE OF CONTENTS

CHAPTER I; INTRODUCTION

1. Statement of the Problem

2. Preliminaries

CHAPTER II: THE L1 +

NORM

CHAPTER III: ORLICZ SPACES AS INTERMEDIATE SPACES

1. Basic Properties of Orlicz Spaces

2. L^ O L^, L^ + L^, and Orlicz Space

3. Monotonic Rearrangement

CHAPTER IV: REARRANGEMENT INVARIANT KOTHE SPACES

1. Normed Spaces

2. Universally Rearrangement Invariant Function

Norms

3. Universal and Universally Rearrangement

Invariant Kothe Spaces

BIBLIOGRAPHY

VITA

Page 1 1 3 9

19 19 21 33 38 38

59

67 72 74

iii

ABSTRACT

Let X^ and

be two Banach spaces contained in a linear

Hausdorff space Y such that the identity mapping of X^(i=l,2)

in Y is continuous. Denote the elements of X^ by f^ and

their norms by IlfII^ . The spaces X^ O

and X^ +

are

Banach spaces under the norms ^ ^ Xj_OX2 = m3X ^ ^ 1 * ^ ^ 2 ^

||f|L ,Y - inf 1 2 f-f1+f2

(||f,||, + ||folL) • A Banach space X C Y

11

1 l

satisfyingX j C X C X j + X2

and ||f||x +x^ s:||f||x *

is called an intermediatespace of X^ and X^ *

an£* X2

Let (A,E,p) be a totally o-finite measure space and let

M(A) be the set of all complex-valued y-measurable functions on

A . Then M(A) is a linear Hausdorff space under convergence in

measure on sets of finite measure. This dissertation is concerned

with determining whether certain classes of norraed Kbthe spaces

(Banach function spaces) are intermediate spaces of L^ = L^(y)

and

L * L (p) .

00

00

It is *proven that

LX. D

00LandL, +X L

0a0re

associate Orlicz spaces and that for every non-trivial Young's

function T there is an equivalent Young's function T' such

that the Orlicz space L ^ , is an intermediate space of L^ and

L^ . The concept of universal function norm is introduced and

it is proven that p is induced by a universal function norm if

and only if

p isa universally rearrangement invariant function iv

norm if and only if p(f) has a representation in terms of f ,

the non-increasing rearrangement of f . The notion of a universal

Kdthe space is presented and it is proven that a KSthe space is

universal if and only if it is universally rearrangement invariant.

It is proven that if A is a universal Kothe space then

L,1 fl L00 C a C L .1 + L00 . Furthermore, if A is normed, in partic-

ular A =

, then there is an equivalent universally rearrange

ment invariant norm p, for which L

is an intermediate space

1

pi

of L, and L

1

00

v

CHAPTER I: INTRODUCTION

1.

Statement of the Problem. Let X^ and X 2 be two Banach

spaces contained in a linear Hausdroff space Y such that the in

jection of X^ (i * 1,2) into Yis continuous. Denote the norm of

by || ||^. The space X^ H X2 is the set of all elements which

are in both X^ and X 2 , and the space X^ + X 2 is the set of all

f e Y of the form f ■ f^ + f2 with f^ e X^ and

e X2 . It is

known that thespaces X^ O X2 and X^ + X2

are Banach spaces under

the norms ||f|fx p x “ maxt ||f • ||f|(2 > and ||fffx + x =

inf{||f1||1 + ||f2II2 : f - £x + f2 , £t e 3.2.1]).

(see [1, p. 165, Prop.

Definition 1.1: A Banach space X C Y satisfying

xxn x2c x C x 1 + x2

and

+ X2 s ««», s »Xln X2

is called an intermediate space of X^ and X2 . Much work has been done on intermediate spaces and the related

topic of interpolation theory. (See [1], [2], [16].) In particular, 1