I'm working on to prove the following: Show that the convergence in the space $\omega$ (space of all sequences with respect to the Fréchet metric) is the coordinate convergence. Any hint is appreciated, thanks.
$\omega$ - space of all sequences with Fréchet metric
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functional-analysis
topological-vector-spaces
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4Are you talking about the metric $d(x,y) = \sum_{n = 1}^{\infty} 2^{-n} \frac{|x_n - y_n|}{1 + |x_{n} - y_{n}|}$? – 2011-02-11
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0Isn't the "natural" metric on the space $\mathbb{R}^\mathbb{N}$ this one: $d(x,x) = 0$ and $d(x,y) = 1/N_{xy}$, where $N_{xy}$ is the first index where $x$ and $y$ differ – 2011-02-11
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0@kahen: Something's wrong with your metric. It'll give the discrete topology on the subspace of sequences with only finitely many non-zero entries, so it can't be a topological vector space b/c scalar multiplication is not continuous. I don't know for sure what you have in mind, but your suggestion only works for pro-discrete things, I guess. – 2011-02-11
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0@matheww: Out of curiosity: who writes $w$ for the space of all sequences and why, I've never seen this before. – 2011-02-11
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0You're correct. I must've been thinking about $\mathbb{N}^\mathbb{N}$ or something. I ought to learn to think things through before writing anything – 2011-02-11
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0@Theo: Some digging on Wikipedia led me to http://en.wikipedia.org/wiki/FK-space#Examples — It seems it's a "TeX-o" for $\omega = \mathbb{C}^\mathbb{N}$ – 2011-02-11
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0@kahen: Thanks! But this really strikes me as more than silly. Or maybe I should reconsider and start writing $\alpha$ for the zero space, in order to single out the alpha and the omega among all sequence spaces... – 2011-02-11
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1Anyway, the original question seems to be pretty well answered by Wikipedia's article on [Fréchet spaces](http://en.wikipedia.org/wiki/Fr%C3%A9chet_space). In particular check out the section on constructing Fréchet spaces and the bit in the Examples section on the family of seminorms on $\mathbb{R}^\omega$ – 2011-02-11
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0@kahen: You're right. The only slightly non-trivial thing to prove is that the $d$ of my first comment actually is a metric (that's why I asked my first question). Why don't you post your last comment as an answer? – 2011-02-11
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0I have seen notation $\omega$ in Wilansky A.: Summability Through Functional Analysis, see [p.3](http://books.google.sk/books?id=vKAMMYuWAAAC&pg=PA3#v=onepage&q&f=false), and in Boos J.: Classical and modern methods in summability. So probably this is usual notation in this area. – 2012-06-14