Posted by J. Holmer on September 15, 2007, 5:23 pm, in reply to "HW3 # 2(b) and 3(d)" Thanks for pointing out the mistake in 3(d). K_1+K_2 is indeed defined as: K_1+K_2 = { x\in X : \exists x_1\in K_1, \exists x_2\in K_2 such that x=x_1+x_2 } The homework 3 pdf file has been updated on the webpage to include this correction. As for 2(b), the problem is correct as stated. d_p is not a map from X\times X \to \mathbb{R}; it is instead a map from (X\times X)\times (X\times X) \to \mathbb{R}. In 2(b), you are supposed to show that for each \epsilon>0, there exists \delta>0 such that for any pair (of pairs) (x_1,x_2) and (y_1,y_2), we have d_p( (x_1,x_2), (y_1,y_2) ) <\delta \implies |\rho(x_1,x_2)-\rho(y_1,y_2)|<\epsilon This is what it means to say that the map \rho is uniformly continuous as a map from the metric space X\times X (endowed with the metric d_p) to the metric space \mathbb{R} (with the usual metric).
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