Norms for a Linear Transformation .... Browder, Lemma 8.4 ....

In summary: Thus, we have $\sum_{ j = 1 }^m ( a_k^j )^2 = |T \mathbf{e}_k|^2 \leqslant \| T \|^2$ as desired. In summary, Lemma 8.4 states that for a linear transformation $T$ and its matrix $A$, the sum of the squares of the entries in each column of $A$ is less than or equal to the squared norm of $T$. This can be demonstrated rigorously using the definitions of matrix and norm.
  • #1
Math Amateur
Gold Member
MHB
3,998
48
I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ...

I need some help in fully understanding Lemma 8.4 ...

Lemma 8.4 reads as follows:
View attachment 9371
View attachment 9372
In the above proof of Lemma 8.4 by Browder we read the following:

" ... ... On the other hand since \(\displaystyle \sum_{ j = 1 }^m ( a_k^j )^2 = \ \mid T e_k \mid \ \le \| T \|^2\) for every \(\displaystyle k, 1 \le k \le n\) ... ... "

My question is as follows:

Can someone please demonstrate rigorously that \(\displaystyle \sum_{ j = 1 }^m ( a_k^j )^2 = \ \mid T e_k \mid \ \le \| T \|^2\) ...
(... ... it seems plausible that \(\displaystyle \sum_{ j = 1 }^m ( a_k^j )^2 = \ \mid T e_k \mid \ \le \| T \|^2\) but how do we demonstrate it rigorously ... ... )
Help will be much appreciated ...

Peter
 

Attachments

  • Browder - 1 - Lemma 8.4 ... PART 1 .. .png
    Browder - 1 - Lemma 8.4 ... PART 1 .. .png
    39.8 KB · Views: 81
  • Browder - 2 - Lemma 8.4 ... PART 2 ... .png
    Browder - 2 - Lemma 8.4 ... PART 2 ... .png
    12.3 KB · Views: 71
Physics news on Phys.org
  • #2
Peter said:
Can someone please demonstrate rigorously that \(\displaystyle \sum_{ j = 1 }^m ( a_k^j )^2 = |T e_k|^2 \le \| T \|^2\) ...
The equality \(\displaystyle \sum_{ j = 1 }^m ( a_k^j )^2 =|T \mathbf{e}_k|^2\) comes from the definition of the matrix of $T$. In fact, $T \mathbf{e}_k = (a_k^1 \mathbf{e}_1,a_k^2 \mathbf{e}_2,\ldots,a_k^m \mathbf{e}_m).$

The inequality $|T \mathbf{e}_k| \leqslant \|T\|$ comes from the definition of $\|T\|$. In fact, $\|T\| = \sup\{|T\mathbf{v}|:\mathbf{v}\in\Bbb{R}^n, |\mathbf{v}|\leqslant1\}$, which implies that $|T\mathbf{v}| \leqslant \|T\|$whenever $|\mathbf{v}|\leqslant1$. But $|\mathbf{e}_k| = 1$, so we can take $\mathbf{v} = \mathbf{e}_k$ to get $|T \mathbf{e}_k| \leqslant \|T\|$.
 

Related to Norms for a Linear Transformation .... Browder, Lemma 8.4 ....

1. What is a linear transformation?

A linear transformation is a function that maps one vector space to another in a way that preserves the basic structure of the vector space, such as addition and scalar multiplication.

2. What are norms for a linear transformation?

Norms for a linear transformation are a way to measure the size or magnitude of a vector in a vector space. They are defined as a function that assigns a non-negative real number to each vector in the space, with the property that the norm of a vector is 0 if and only if the vector is the zero vector.

3. What is Browder's lemma 8.4?

Browder's lemma 8.4 is a mathematical theorem that states that if a linear transformation between two vector spaces satisfies certain conditions, then the transformation is continuous. This lemma is often used in the study of functional analysis and operator theory.

4. How are norms for a linear transformation used in mathematics?

Norms for a linear transformation are used in mathematics to measure the size or magnitude of vectors in a vector space. They are also used to define metrics and distances between vectors, which are important tools in many areas of mathematics such as analysis, geometry, and optimization.

5. Can norms for a linear transformation be generalized to other mathematical structures?

Yes, norms for a linear transformation can be generalized to other mathematical structures, such as normed spaces, Banach spaces, and Hilbert spaces. In these settings, the concept of a norm is extended to include functions, sequences, and other objects, and plays a crucial role in the study of these mathematical structures.

Similar threads

  • Linear and Abstract Algebra
Replies
2
Views
1K
  • Topology and Analysis
Replies
2
Views
1K
  • Linear and Abstract Algebra
Replies
8
Views
1K
  • Linear and Abstract Algebra
Replies
8
Views
1K
Replies
4
Views
1K
  • Linear and Abstract Algebra
Replies
4
Views
1K
  • Topology and Analysis
Replies
2
Views
1K
  • Topology and Analysis
Replies
13
Views
1K
Replies
2
Views
1K
Back
Top