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I am reading Michael Field's book: "Essential Real Analysis" ... ...
I am currently reading Chapter 9: Differential Calculus in $$\mathbb{R}^m$$ 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:
$$ \text{sup}_{ \| u \| = 1} ( \| Au \| + \| Bu \| ) \le \text{sup}_{ \| u \| = 1} ( \| Au \| ) +\text{sup}_{ \| u \| = 1} ( \| Bu \| ) $$
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 $$\mathbb{R}^m$$ 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:
$$ \text{sup}_{ \| u \| = 1} ( \| Au \| + \| Bu \| ) \le \text{sup}_{ \| u \| = 1} ( \| Au \| ) +\text{sup}_{ \| u \| = 1} ( \| Bu \| ) $$
Can someone please explain why this is true ... surely the above is a strict equality ...
Help will be much appreciated ...
Peter