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Let $(X,\rVert{\cdot}\lVert)$ be a normed vector space. Clearly there is a canonical way to induce a metric (and so a topology) on $X$ by defining $d(x,y)=\rVert x-y\rVert$.

Are there other metrics induced by $\rVert{\cdot}\lVert$? If does exist another metric $d_1\neq d$ induced by $\rVert{\cdot}\lVert$, are $d$ and $d_1$ equivalent?

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    $\begingroup$ What does "equivalent" mean here? $\endgroup$
    – 23rd
    May 2 '13 at 8:49
  • $\begingroup$ I mean that the induced topology (with $\epsilon$-balls) is the same. $\endgroup$
    – Dubious
    May 2 '13 at 8:50
  • $\begingroup$ I think metric induced by $||.||$ is defined as $||x-y||$, can you explain what do you mean by 'metric induced by $||. ||$' ? $\endgroup$
    – pritam
    May 2 '13 at 8:51
  • $\begingroup$ For example a metric $d_1=f(\rVert x-y\lVert)$ where $f$ is a generic function. $\endgroup$
    – Dubious
    May 2 '13 at 8:53
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    $\begingroup$ Then there are lots of such metrics. For example, $d_1=2d$. $\endgroup$
    – 23rd
    May 2 '13 at 8:54
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Even if the question is loosely formulated, I would say that the answer is no. I'm thinking at the following examples: in the vector space $\mathbb{R}$, consider the usual absolute value $\lvert\cdot\rvert$, which indeed is a norm and induces the ordinary topology of the real line. We can define a metric on $\mathbb{R}$ in terms of $\lvert\cdot\rvert$ as follows: $$d_1(x, y)=\lvert \arctan(x)-\arctan(y)\rvert,$$ or as follows: $$d_2(x, y)=\frac{\lvert x-y\rvert}{1+\lvert x-y\rvert}$$.

Both are "metrics induced by $\lvert \cdot\rvert$" in some vague sense, and neither of them is metrically equivalent to $\lvert \cdot\rvert$, because both make $\mathbb{R}$ into a bounded space.

P.S. : I had not noticed that you consider "equivalent" two metrics which induce the same topology. Then $d_1$ and $d_2$ above do not qualify as counterexamples because they do induce the ordinary topology of the line. To obtain a counterexample in this case you can take the $\text{signum}$ function $$\text{signum}(s)=\begin{cases} 1& s>0 \\ 0 & s=0 \\ -1& s<0\end{cases}$$ and define $$d_3(x, y)=\text{signum}(\lvert x-y\rvert).$$ You get the discrete metric on $\mathbb{R}$, which obviously does not induce the same topology on $\mathbb{R}$ as $\lvert\cdot\rvert$.

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