Sequence space s (Functional analysis, Kreyszig)

Question

I'm reading Kreyszig's functional analysis book, in which are the following example:

$\textbf{1.2-1 Sequence space s.}$ This space consists of the set of all (bounded or unbounded) sequences of complex numbers and the metric $d$ defined by

$$ {d(x,y) = \sum_{j=1}^{\infty} \frac{1}{2^j}\frac{|\epsilon_{j}-\eta_{j}|}{1 + |\epsilon_{j}-\eta_{j}|}}, $$

where $x = (\epsilon_{j})$ and $y = (\eta_{j})$.

Could you give me more references about this sequence space (I didn't find any on the Internet by this name), or about metric spaces like this (namely, defined by series)?

Note: Much more, better, because I'm a beginner in functional analysis.

Answer 1

I will expand a bit on my comment.

This space isn't as interesting as $\ell^p$, $\ell^\infty$, and others because it is not a normed vector space. Elementary functional analysis concerns itself with linear functionals and operators as well as with the representation of vectors inside infinite dimensional vector spaces. Its way easier to build results in these directions when the available notion of distance has some compatibility with the vector space structure.

To prove my point, if $d$ came from a norm by $d(x,y) = \lvert\lvert x-y \rvert \rvert$, then $d(ax,ax) = \lvert a \rvert \lvert\lvert x-y \rvert \rvert$. This doesn't happen in our case. Indeed, if we let $x = (1,1,...)$ and $y = (0,0,...)$, then $d(x,y) = \sum_{i=1}^\infty \frac{1}{2^i} \frac{1}{1 + 1} = 1/2$. At the same time, $d(3x,3y) = \sum_{i=1}^\infty \frac{1}{2^i} \frac{3}{4} = 3/4 \neq 3/2$.