Every finite n-dimensional Banach space are equivalent

Question

I am in need of some clarification here.

My doubts are highlighted in yellow in the image above.

I have 2 questions:

What does $\left \| \vec{w_{j}} \right \|$ stands for? Why is $\left \| \phi\left ( \vec{x} \right ) \right \| \leq \frac{k}{c}\left | \vec{x} \right |$?

Explanation would be very much appreciated.

Thanks in advance.

Answer 1

$w_j$ is a vector in $W$, and $\|w_j\|$ is the norm of that vector. $\max_j\|w_j\|$ is the maximum among the values $\|w_1\|,\|w_2\|,\dots,\|w_n\|$. That is, it is the maximum among the $\|w_j\|$, where $j$ goes from $1$ to $n$.

For the second: by Lemma 5.4, there exists a $c$ such that $$ |x| = |x_1 v_1 + \cdots + x_n v_n| \geq c(|x_1| + \cdots + |x_n|) $$ which is to say that $$ |x_1| + \cdots + |x_n| \leq \frac 1c |x| $$

and it is this $c$ which is used in the inequality.

Answer 2

First question: $\|w_j\|$ is the norm of $w_j $.

Second question: by Lemma 5.4.

Answer 3

First, since $\| \cdot \|$ is the norm on $W$, $k$ is defined as the maximum of the $W$-norms of the $W$-basis $\{w_1, ..., w_n\}$. That is, $k = \max\{\|w_j\|: j = 1, ..., n\}$.

Second, although I'm not sure what Lemma 5.4 is, I suspect you can deduce something to the extent that if $x = x^1v_1 + \cdots + x^n v_n$, then $|x|_V \leq \frac{1}{c} \sum\limits_{i=1}^n |x^i|$, where you should realize that I have added a subscript $V$ where I'm using the $V$ norm as opposed to the absolute value of the coefficients.

Answer 4

For the second question apply Lemma $5.4$ with $\alpha_i=x_i$.