# Operator Norm and Distance Function ... Browder, Proposition 8.6 ...

#### Peter

##### Well-known member
MHB Site Helper
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 the concepts in Proposition 8.6 ...

In the above proposition, Browder defines the distance function \rho (S, T) as follows:

$$\displaystyle \rho (S, T) = \| S - T \|$$

... but just some basic questions ...

How do we define $$\displaystyle -T$$?

Is $$\displaystyle -T = ( -1) T$$?

Is $$\displaystyle \| -T \| = \| T \|$$?

A simple example that shows the way things work as I see it follows:

Consider $$\displaystyle T: \mathbb{R}^2 \to \mathbb{R}^2$$

Let $$\displaystyle T(x,y) = ( x - y, 2y )$$

... then ...

$$\displaystyle - T (x,y) = (-1) T(x,y) = ( -x + y, -2y)$$

and then it follows that ...

$$\displaystyle \| T(x,y) \| = \| ( x - y, 2y ) \| = \sqrt{ (x - y)^2 + (2y)^2 }$$

and ...

$$\displaystyle \| -T(x,y) \| = \| ( -x + y, -2y) \| = \sqrt{ (-x + y)^2 + (-2y)^2 } = \| T(x,y) \|$$

Is the above example correct?

Peter

#### LCKurtz

##### New member
Yes, it looks like you have everything correct.

#### Peter

##### Well-known member
MHB Site Helper
Yes, it looks like you have everything correct.

Thanks LCKurtz

Peter