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I am reading Michael Field's book: "Essential Real Analysis" ... ...

I am currently reading Chapter 9: Differential Calculus in \(\displaystyle \mathbb{R}^m\) and am specifically focused on Section 9.2.1 Normed Vector Spaces of Linear Maps ...

I need some help in fully understanding Theorem 9.2.9 (3) ... Theorem 9.2.9 (3) reads as follows:

View attachment 9375

View attachment 9376

In the proof of Theorem 9.2.9 (3) Field asserts the following:

\(\displaystyle \text{sup}_{ \| u \| = 1} ( \| Au \| + \| Bu \| ) \le \text{sup}_{ \| u \| = 1} ( \| Au \| ) +\text{sup}_{ \| u \| = 1} ( \| Bu \| ) \)

Can someone please explain why this is true ... surely the above is a strict equality ...

Help will be much appreciated ...

Peter

I am currently reading Chapter 9: Differential Calculus in \(\displaystyle \mathbb{R}^m\) and am specifically focused on Section 9.2.1 Normed Vector Spaces of Linear Maps ...

I need some help in fully understanding Theorem 9.2.9 (3) ... Theorem 9.2.9 (3) reads as follows:

View attachment 9375

View attachment 9376

In the proof of Theorem 9.2.9 (3) Field asserts the following:

\(\displaystyle \text{sup}_{ \| u \| = 1} ( \| Au \| + \| Bu \| ) \le \text{sup}_{ \| u \| = 1} ( \| Au \| ) +\text{sup}_{ \| u \| = 1} ( \| Bu \| ) \)

Can someone please explain why this is true ... surely the above is a strict equality ...

Help will be much appreciated ...

Peter