Consider the metric space $(\mathbb{R}^{\mathbb{N}},d)$ where for $x,y\in\mathbb{R}^\mathbb{N}$ $$ d(x,y) = \sum_{n=1}^{\infty} 2^{- n} \frac{\bigvee_{k\leq n}\left|x_k-y_k\right|}{1 + \bigvee_{k\leq n} \left|x_k-y_k\right|}.$$ I am wondering if that metric is well-known in the context of the infinite-dimensional space $\mathbb{R}^\mathbb{N}$ and whether it has a name. Does it make the space $\mathbb{R}^{\mathbb N}$ complete?
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2What is $\bigvee_{n = 1}^{\infty} \left| x_n - y_n \right|$? – Michael Greinecker Nov 26 '12 at 09:49
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@MichaelGreinecker $\bigvee_n$ usually denotes $\sup_n$. And learner: What do you mean by $\mathbb R^\infty$? All sequences? Or only the finally zero ones? – martini Nov 26 '12 at 09:52
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@learner It is somewhat standard. But in that case, you do not really get a metric. The supremum can be infinite. – Michael Greinecker Nov 26 '12 at 09:53
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I apologize Michael, I made a mistake in the definition. I will correct it. – Learner Nov 26 '12 at 09:56
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2This seems like a near-duplicate of http://math.stackexchange.com/questions/21552/omega-space-of-all-sequences-with-frechet-metric/21633 – kahen Nov 26 '12 at 10:13
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Yes, it seem like it metrizes the product topology (which could be done easier). – Michael Greinecker Nov 26 '12 at 12:40