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

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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.
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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 ...

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

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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 $|T \mathbf{e}_k| \leqslant \|T\|$.

## 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.

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