What is Banach space?

Example 5.20 (The operator norm of integration)

On $\bigl(C(\intcc01), \norm\cdot_\infty\bigr)$, let $T(f)(x) = \int_0^x f(t)\,\dd t$ (an endomorphism: $T(f)$ is continuous). Run Method 5.9. Upper bound:

so $\vertiii T \leq 1$. Witness: $f \equiv 1$ gives $T(f)(x) = x$ and $\norm{T(f)}_\infty = 1 = \norm f_\infty$: attained, $\vertiii T = 1$. But note $\vertiii{T^2} = \frac12 \neq \vertiii T^2$: indeed $T^2(f)(x) = \int_0^x(x - t)f(t)\dd t$ has $\abs{T^2(f)(x)} \leq \frac{x^2}2\norm f_\infty$, attained again at $f \equiv 1$; and generally $\vertiii{T^n} = \frac1{n!}$ — the submultiplicative bound $\vertiii T^n = 1$ is off by a factorial. This is exactly the phenomenon the iterate trick of Chapter 4’s weekend problem converts into global solvability of linear differential equations.

Example 5.22 (Matrix exponential, first contact)

The matrix exponential: $\mathcal{M}_n(K)$ with any submultiplicative norm ($\vertiii{AB} \leq \vertiii A \vertiii B$) is Banach (finite dimension). Then, for every $A$,

converges absolutely ($\vertiii{A^k/k!} \leq \vertiii A^k /k!$, summable): well defined. Chapter 16 exploits it systematically.