Linear transformations between normed linear spaces

Click For Summary
SUMMARY

This discussion focuses on linear transformations between normed linear spaces (NLS) and the definition of the operator norm ||T||. Specifically, it addresses the equivalence of the definitions ||T||: sup{||T|| : ||x||<=1} and ||T||: sup{||T|| : ||x||=1} for non-zero spaces X. The participant expresses frustration over the lack of clarity in textbooks regarding this equivalence, despite intuitively understanding it through the bounded nature of T and the manipulation of functions within a closed unit sphere.

PREREQUISITES
  • Understanding of linear transformations in mathematics
  • Familiarity with normed linear spaces (NLS)
  • Knowledge of operator norms and their properties
  • Basic concepts of bounded functions in functional analysis
NEXT STEPS
  • Study the properties of operator norms in normed linear spaces
  • Explore the concept of bounded linear transformations in functional analysis
  • Learn about the closed unit sphere in normed spaces
  • Investigate the relationship between suprema and infima in mathematical analysis
USEFUL FOR

Mathematicians, students of functional analysis, and anyone studying linear algebra who seeks to deepen their understanding of linear transformations and operator norms in normed linear spaces.

Ant farm
Messages
19
Reaction score
0
Hi, ok I'm working with linear transformations between normed linear spaces (nls)

if T :X -> Y nls's is a linear transformation, we define the norm of T, ||T||: sup{||T|| : ||x||<=1}

I want to show that for X not = {0}
||T||: sup{||T|| : ||x|| = 1} frustratingly the books all assume that this step is obvious... I don't see how.

Intuitively I can see that is true, using the fact that (I think) T is a bounded function, and we can manipulate things to make the whole function be contained within a closed unit sphere...
Have always had a mental block with inf's and sups...don't know why...
 
Physics news on Phys.org
Work with x/||x||. It will be obvious.
 

Similar threads

High School Linearization of a function error viewed with differentials
  • · Replies 14 ·
Replies
14
Views
3K
Undergrad On (real) entire functions and the identity theorem
  • · Replies 2 ·
Replies
2
Views
4K
The correct way to write the range of a linear transformation
  • · Replies 11 ·
Replies
11
Views
2K
Undergrad Confused about small detail in rank-nullity theorem
  • · Replies 1 ·
Replies
1
Views
3K
Graduate How do I know "what Fourier transform" to use?
  • · Replies 3 ·
Replies
3
Views
2K
Linear operator in 2x2 complex vector space
  • · Replies 18 ·
Replies
18
Views
3K
Undergrad Lipschitz continuity of vector-valued function
  • · Replies 3 ·
Replies
3
Views
4K
Graduate What is the meaning of the norm of a linear functional?
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Wronskian: null space equals the span of independent functions
  • · Replies 23 ·
Replies
23
Views
2K
Question about linear transformations
  • · Replies 3 ·
Replies
3
Views
2K