这是一份warwick华威大学MA132-10的成功案例

Let $F \subset \mathbb{R}^{n}$ and suppose that $f: F \rightarrow \mathbb{R}^{m}$ satisfies a Hölder condition
$$
|f(x)-f(y)| \leqslant c|x-y|^{\alpha} \quad(x, y \in F) .
$$
Then $\operatorname{dim}{H} f(F) \leqslant(1 / \alpha) \operatorname{dim}{H} F$.
Proof. If $s>\operatorname{dim}{H} F$ then by Proposition $2.2 \mathcal{H}^{s / \alpha}(f(F)) \leqslant c^{s / \alpha} \mathcal{H}^{s}(F)=0$, implying that $\operatorname{dim}{\mathrm{H}} f(F) \leqslant s / \alpha$ for all $s>\operatorname{dim}_{\mathrm{H}} F$.

MA132-10 COURSE NOTES ：

Fundamental to most definitions of dimension is the idea of ‘measurement at scale $\delta$. For each $\delta$, we measure a set in a way that ignores irregularities of size less than $\delta$, and we see how these measurements behave as $\delta \rightarrow 0$. For example, if $F$ is a plane curve, then our measurement, $M_{\delta}(F)$, might be the number of steps required by a pair of dividers set at length $\delta$ to traverse $F$. A dimension of $F$ is then determined by the power law (if any) obeyed by $M_{\delta}(F)$ as $\delta \rightarrow 0$. If
$$
M_{\delta}(F) \sim c \delta^{-s}
$$
for constants $c$ and $s$, we might say that $F$ has ‘divider dimension’ $s$, with $c$ regarded as the ‘ $s$-dimensional length’ of $F$. Taking logarithms
$$
\log M_{\delta}(F) \simeq \log c-s \log \delta
$$
in the sense that the difference of the two sides tends to 0 with $\delta$, and
$$
s=\lim {\delta \rightarrow 0} \frac{\log M{\delta}(F)}{-\log \delta}
$$