等价/强等价度量保持的性质

hbghlyj

对于一个同胚 $f:(0,1)→ℝ$, 可以由$ℝ$上的通常度量$d(x,y)=\abs{x-y}$导出$(0,1)$上的度量:$$\tilde d(x,y)=d\left(f(x),f(y)\right)$$$(0,1)$的这两个度量是等价的, 却不是强等价的.

Equivalence of metrics

A metric that is strongly equivalent to a complete metric is also complete; the same is not true of equivalent metrics because homeomorphisms do not preserve completeness. For example, since \((0,1)\) and \(\mathbb R\) are homeomorphic, the homeomorphism induces a metric on \((0,1)\) which is complete because \(\mathbb R\) is, and generates the same topology as the usual one, yet \((0,1)\) with the usual metric is not complete, because the sequence \((2^{-n})_{n\in\mathbb N}\) is Cauchy but not convergent. (It is not Cauchy in the induced metric.)

$\left((0,1),\tilde d\right)$ 是完备度量空间. 例如\((2^{-n})_{n\in\mathbb N}\)是$\left((0,1),d\right)$中的柯西列, 但不是$\left((0,1),\tilde d\right)$中的柯西列. 设$(x_n)_{n\in\mathbb N}$为$\left((0,1),\tilde d\right)$中的任意一个柯西列, 则\(\big(f(x_n)\big)_{n\in\mathbb N}\)是$ℝ$中的柯西列, 故收敛到一个实数$c$, 则\((x_n)_{n\in\mathbb N}\)在$\left((0,1),\tilde d\right)$中收敛到$f^{-1}(c)$,但在$\big((0,1),d\big)$不收敛.

The continuity of a function is preserved if either the domain or range is remetrized by an equivalent metric, but uniform continuity is preserved only by strongly equivalent metrics.

上面的例子中,$f$在$d$下是连续但不一致连续的,在$\tilde d$下却是一致连续的.

hbghlyj

The differentiability of a function $f : U → V$, for $V$ a normed space and $U$ a subset of a normed space, is preserved if either the domain or range is renormed by a strongly equivalent norm.

这个如何证明呢