Function properties preserved under a change of metrics?

Question

Consider two metric spaces $(X, d_X)$ and $(Y,d_Y)$, with $d_X$ and $d_Y$ induced by two norms. Suppose function $f: X → Y$ has or lacks some properties, such as:

1. Continuity
2. Uniform Continuity
3. Lipschitz Continuity with constant K
4. Contraction mapping
5. Differentiabiliy

Now suppose we replace $d_X$ and $d_Y$ by metrics $d_{X}'$ and $d_{Y}'$, where $d_{X}'$ and $d_{Y}'$ are induced by norms that are equivalent to those that induced $d_X$ and $d_Y$.

Does the function $f$ possess the same properties (1-5) with these new metrics?

I found it surprisingly hard to figure out if that is the case - most sources I've seen just prove that all norms are equivalent in a finite-dimensional linear space, without really showing why we would care for norm equivalence at all.

EDIT: It doesn't work for Lipschitz Continuity as described here.

Answer 1

Equivalent norms induce the same open sets — so $(X, d_X)$ and $(X, d_X')$ have the same underlying topological space.

So, anything that can be expressed in terms of topology respects equivalence of norms. For example:

• Continuity of a function has a simple description in terms of open sets,
• Uniform continuity can be defined on topological vector spaces.