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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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What is $\bigvee_{n = 1}^{\infty} \left| x_n - y_n \right|$? – Michael Greinecker Nov 26 '12 at 9:49
@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 9:52
@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 9:53
I apologize Michael, I made a mistake in the definition. I will correct it. – Learner Nov 26 '12 at 9:56
This seems like a near-duplicate of… – kahen Nov 26 '12 at 10:13

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