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Mathematics · 术语表

什么是 Banach space？

Definition 5.19 University Mathematics — Year 2 · 第 5 章 — Normed Vector Spaces

A Banach space is a complete normed space. Examples: every finite-dimensional normed space (Theorem 5.13); $\bigl(C(\intcc{a}{b}), \norm\cdot_\infty\bigr)$ (Theorem 4.9); $\mathcal{L}_c(E, F)$ for $F$ Banach (same proof pattern as for continuous functions). Non-example: $\bigl(C(\intcc{0}{1}), \norm\cdot_1\bigr)$ (Exercise 5.7).

示例

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:

\abs{T(f)(x)} \leq \int_0^x\abs f \leq x\,\norm f_\infty \leq \norm f_\infty ,

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 $A$ ,

\eu^A = \sum_{k=0}^{\infty} \frac{A^k}{k!}

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

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