Docsity
Docsity
Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity

Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips
Guidelines and tips
Sell on Docsity
Sell on Docsity
Docsity AI
Docsity AI
Log inSign up

Prepare for your exams

Study with the several resources on Docsity

Find documents

Prepare for your exams with the study notes shared by other students like you on Docsity

Search for your university

Find the specific documents for your university's exams

Docsity AINEW

Summarize your documents, ask them questions, convert them into quizzes and concept maps

Explore questions

Clear up your doubts by reading the answers to questions asked by your fellow students

Earn points to download

Earn points by helping other students or get them with a premium plan

Share documents
download point icon20 Points

For each uploaded document

Answer questions
download point icon5 Points

For each given answer (max 1 per day)

All the ways to get free points
Get points immediately

Choose a premium plan with all the points you need

Study Opportunities

Choose your next study program

Get in touch with the best universities in the world. Search through thousands of universities and official partners

Community

Ask the community

Ask the community for help and clear up your study doubts

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

Homework 5 for Math 554: Normed Linear Spaces - Prof. Eduard-Wilhelm Kirr, Assignments of Linear Algebra

University of Illinois - Urbana-ChampaignLinear Algebra
Prof. Eduard-Wilhelm Kirr

Four problems related to normed spaces, linear and compact operators, adjoint operators, and the fredholm alternative. The problems require understanding of functional analysis, specifically the properties of normed spaces, compact operators, and the adjoint operator. The fourth problem also involves the fredholm alternative and the solvability of certain equations.

Typology: Assignments

Pre 2010

Uploaded on 03/11/2009

koofers-user-vqb
koofers-user-vqb ๐Ÿ‡บ๐Ÿ‡ธ

10 documents

1 / 1

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
Math 554
Homework 5
Due November 14, 2008, before class
Unless otherwise stated, in what follows Xand Ywill be normed spaces over
the same K=R,or C.
Problem I If L:X7โ†’ Yis linear and continuous then the adjoint operator
is continuous and kLโˆ—k=kLk.
Problem II If A:X7โ†’ Xis linear and compact then yโˆ—โˆˆRange(Aโˆ—โˆ’Id)
implies ker(Aโˆ’Id)โŠ†ker yโˆ—.(This is the necessity part of Lemma 2).
Problem III Assume Xis a reflexive Banach space and A:X7โ†’ Xis
linear and compact. Show directly, using Fredholm alternative, that
yโˆ—โˆˆRange(Aโˆ—โˆ’Id) is equivalent to ker(Aโˆ’Id)โŠ†ker yโˆ—.
Problem IV Assume A:X7โ†’ Xis linear and compact. Show that the
equations:
Ax โˆ’x=y, y โˆˆX(1)
Aโˆ—xโˆ—โˆ’xโˆ—=yโˆ—, yโˆ—โˆˆXโˆ—,(2)
are either both uniquely solvable or, if they are solvable they have the
same number of (finitely many) independent solutions. In the latter
case what condition on yrespectively yโˆ—are equivalent with the solv-
ability of (1) respectively of (2).

Partial preview of the text

Download Homework 5 for Math 554: Normed Linear Spaces - Prof. Eduard-Wilhelm Kirr and more Assignments Linear Algebra in PDF only on Docsity!

Math 554

Homework 5

Due November 14, 2008, before class

Unless otherwise stated, in what follows X and Y will be normed spaces over the same K = R, or C.

Problem I If L : X 7 โ†’ Y is linear and continuous then the adjoint operator is continuous and โ€–Lโˆ—โ€– = โ€–Lโ€–.

Problem II If A : X 7 โ†’ X is linear and compact then yโˆ—^ โˆˆ Range(Aโˆ—^ โˆ’ Id) implies ker(A โˆ’ Id) โŠ† ker yโˆ—. (This is the necessity part of Lemma 2).

Problem III Assume X is a reflexive Banach space and A : X 7 โ†’ X is linear and compact. Show directly, using Fredholm alternative, that yโˆ—^ โˆˆ Range(Aโˆ—^ โˆ’ Id) is equivalent to ker(A โˆ’ Id) โŠ† ker yโˆ—.

Problem IV Assume A : X 7 โ†’ X is linear and compact. Show that the equations:

Ax โˆ’ x = y, y โˆˆ X (1) Aโˆ—xโˆ—^ โˆ’ xโˆ—^ = yโˆ—, yโˆ—^ โˆˆ Xโˆ—, (2)

are either both uniquely solvable or, if they are solvable they have the same number of (finitely many) independent solutions. In the latter case what condition on y respectively yโˆ—^ are equivalent with the solv- ability of (1) respectively of (2).