Question about Normed Linear Spaces

  • Thread starter Oster
  • Start date
  • Tags
    Linear
In summary, the statement discusses a finite dimensional vector space V with a basis {ei} and a norm defined on it. The goal is to prove that the norm of any vector x is greater than or equal to the norm of the basis vector xiei for any fixed i. However, there is uncertainty about the correctness of this result and the independence of the basis vectors may play a role in the proof. An example is given to show that the statement is not always true.
  • #1
Oster
85
0
Statement: V is a finite dimensional vector space with basis {ei} (i goes from 1 to n). V has a norm || || defined on it(not necessarily induced by an inner product). Let x=Ʃxiei belong to V. I want to show that ||x|| ≥ ||xiei|| for any fixed i.

I'm not entirely sure this result is correct. But i remember seeing something similar in a text a while ago.
I know all the properties of a norm but I'm not sure how to proceed. I don't know how the independence of the basis vectors will fit into the proof.
 
Physics news on Phys.org
  • #2
It isn't true. In ##R^2## consider the basis ##e_1=<1,0>,\ e_2=<-1,.1>##. Then let ##x=1e_1+1e_2=<0,.1>##. Then ##\|x\|=.1<1\|e_1\|##.
 
  • #3
Thank you!
 

1. What is a normed linear space?

A normed linear space is a mathematical structure that combines the concepts of a vector space and a metric space. It is a vector space equipped with a norm, which is a function that assigns a non-negative value to each vector, satisfying certain properties such as the triangle inequality and the non-negativity of the norm of the zero vector.

2. What is the purpose of studying normed linear spaces?

Normed linear spaces are important in functional analysis, which is a branch of mathematics that studies vector spaces and their linear transformations. They provide a framework for studying the properties and behaviors of vectors and their operations, as well as the convergence and continuity of functions defined on these spaces.

3. How are normed linear spaces different from other types of vector spaces?

A normed linear space differs from a general vector space in that it has an extra structure, the norm, which allows for the definition of concepts such as distance and convergence. It is also more restrictive than a normed vector space, which requires the norm to satisfy the additional property of homogeneity.

4. What are some examples of normed linear spaces?

Some commonly studied examples of normed linear spaces include Euclidean spaces, Banach spaces, and Hilbert spaces. Other examples include function spaces such as Lp spaces and spaces of continuous functions, as well as sequence spaces such as l^p spaces and the space of convergent sequences.

5. How are normed linear spaces used in real-world applications?

Normed linear spaces have various applications in fields such as physics, engineering, and economics. They are used to model physical quantities such as velocity and acceleration, as well as to solve optimization problems in economics. In engineering, they are used to study the convergence and stability of numerical methods, and in data analysis, they are used to measure the similarity between data points.

Similar threads

V Space With Norm $||*||$ - Fourier Series
    • Calculus and Beyond Homework Help
Replies
26
Views
1K
Null space of dual map of T = annihilator of range T
    • Calculus and Beyond Homework Help
Replies
0
Views
440
Does each norm on vector space become discontinuous when restricted to S^1?
    • Calculus and Beyond Homework Help
Replies
1
Views
566
Given specific v, dimension of subspace of L(V, W) where Tv=0?
    • Calculus and Beyond Homework Help
Replies
14
Views
577
Finding a complementary subspace ##U## | Linear Algebra
    • Calculus and Beyond Homework Help
Replies
4
Views
1K
Vector space of functions defined by a condition
    • Calculus and Beyond Homework Help
2
Replies
58
Views
3K
Prove that the linear space is infinite dimensional
    • Calculus and Beyond Homework Help
Replies
18
Views
1K
Bounded operators on Hilbert spaces
    • Calculus and Beyond Homework Help
2
Replies
43
Views
3K
A quite verbal proof that if V is finite dimensional then S is also....
    • Calculus and Beyond Homework Help
Replies
34
Views
2K
Optimization: Dual for L1 norm minimization with equality constraint
    • Calculus and Beyond Homework Help
Replies
1
Views
544

Hot Threads

Recent Insights

Back
Top