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I have a problem understanding the example 2.2-8 from Kreyszig's "Introductory Functional Analysis with Applications". The example shows a metric $d(x,y)$ on the space $s$ - set of all (bounded or unbounded) sequences of complex numbers: $$ d(x,y) = \sum_{j=1}^\infty \frac{1}{2^j}\frac{|\zeta_j - \eta_j|}{1+|\zeta_j - \eta_j|} $$

Question: Why exactly can't this metric be obtained from a norm?

Konstantin
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    If this came from a norm, $|| \cdot ||$, you would have $||cx|| = |c| \cdot ||x||$ for every complex number $c$ and every sequence $x$. – D_S Mar 10 '17 at 19:23

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If a metric on a vector space comes from a norm, the norm must be $\lVert x\rVert=d(x,0)$. In the example you cited, the metric is actually bounded, and a metric that comes from a norm can't possibly be bounded, because of homogeneity (unless the space is zero-dimensional, at least when we're talking about standard norms over reals or complex numbers).

But even unbounded metrics need not come from norms. Probably one of the most natural examples is the space of $C^\infty$ functions on the interval $[0,1]$, or the space $\ell^p$ for $p\in (0,1)$.

tomasz
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