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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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    Yes, it seem like it metrizes the product topology (which could be done easier).2012-11-26

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