A proof of a general property of norms

Click For Summary
SUMMARY

The discussion revolves around proving the inequality | ||x|| - ||y|| | <= ||x - y|| for any norm on any vector space. The initial poster expresses confusion about how to approach the proof, indicating a lack of understanding of fundamental properties of norms. A key insight provided by a respondent is that this inequality follows from the triangle inequality, specifically by considering the expression ||x - y + y||.

PREREQUISITES
  • Understanding of vector spaces
  • Familiarity with norms and their properties
  • Knowledge of the triangle inequality in mathematics
  • Basic proof techniques in mathematical analysis
NEXT STEPS
  • Study the properties of norms in vector spaces
  • Learn about the triangle inequality and its applications
  • Explore different types of norms, such as Lp norms
  • Practice proving inequalities involving norms
USEFUL FOR

Mathematics students, researchers in functional analysis, and anyone interested in understanding the properties of norms in vector spaces.

gucci1
Messages
12
Reaction score
0
So I have been asked to prove a result that is supposedly valid for any norm on any vector space. The statement to prove is: | ||x|| - ||y|| | <= ||x - y||

The problem is, I have no idea where to start with this proof. Maybe I'm missing some fundamental property of norms, but it seems that having to prove this for any norm on any vector space rules out any definitions that I might try to apply :-/

Sorry for not having anything to go on here, but I'm lost. Thank you all very much for any help you might be able to offer!
 
Physics news on Phys.org
This follows from the triangle inequality. Consider the quantity $\| x-y+y \|$.
 

Similar threads

Graduate Proof of the inequality of a reduced basis
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Proof that if T is Hermitian, eigenvectors form an orthonormal basis
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Shouldn't this definition of a metric include a square root?
  • · Replies 8 ·
Replies
8
Views
2K
Graduate Normed linear space vs inner product space and more
  • · Replies 15 ·
Replies
15
Views
11K
Undergrad Uniqueness of reduced row echelon form (proof verification)
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Proof concerning the Four Fundamental Spaces
  • · Replies 14 ·
Replies
14
Views
3K
Undergrad Proving 2-Norm of A: Understanding the Relationship Between u and v
  • · Replies 6 ·
Replies
6
Views
5K
Undergrad Finding the orthogonal projection of a vector without an orthogonal basis
  • · Replies 3 ·
Replies
3
Views
4K
Undergrad ##L^2## square integrable function Hilbert space
  • · Replies 11 ·
Replies
11
Views
3K
High School Does the L2 norm of a vector destroy all directional info?
  • · Replies 6 ·
Replies
6
Views
2K