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Claim: $R^\infty \text{ is not a Banach space when equipped with its natural product topology}$ I need help proving this 'obvious' claim. I just got acquainted with a definition of a product topology and the concept does not seem to be easy to work with. How would I even go about showing whether $R^\infty$ is metrizable?

Edit: Intuitively, I know that $d(x,y)=\sum_{j=1}^{\infty}\frac{1}{2^j} \frac{|x_j-y_j|}{1+|x_j-y_j|}$ should prove that $R^{\infty}$ is metrizable.

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    I am asking for a proof of a claim. Sorry for omitting it, I will edit2012-01-23

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Hint: any basic neighbourhood of $0$ in the product topology will contain a straight line through $0$.

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    if $\{tv,t \in \mathbb{R}\} \subset B(0,1)$ for some $v \in \mathbb{R}^{\infty}$, just take t > 1/||v|| to conclude there's no line.2012-01-24