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In a normed linear space $(X,\|\cdot\|)$, by definition we have $\forall x,y \in X$, $$\|x+y\| \leq \|x\|+\|y\|.$$ My question is, is it true (or does the definition of norm imply) that for a sequence $\{x_n\} \in X$, $$\left\|\sum_{i=1}^{\infty}x_i\right\| \leq \sum_{i=1}^{\infty}\|x_i\|?$$ What about the more general case of uncountable summation?

For example, does the Minkowski inequality hold for a more general case of countable or even uncountable summation?


Intuitively, I think all above are true, but I'm still not quite confident..:(

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    $\begingroup$ Definitely holds for finite sums - try induction on $n$. $\endgroup$ – stochasticboy321 Dec 5 '15 at 19:01
  • $\begingroup$ How do you perform an uncountable sum? Which order are you taking for computing the sum $\endgroup$ – mathcounterexamples.net Dec 5 '15 at 19:02
  • $\begingroup$ @stochasticboy321 Sorry, a typo. Changed it. I guess It's true for infinite summation because $||x||$ is a continuous function? $\endgroup$ – 826313315 Dec 5 '15 at 21:16
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For finite sums, it works. For non-finite sums, as long as the series is defined (which is slightly more tricky for nets than it is for sequences), yes. You would take the inequality for the partial sums and then take the limit.

The caveat, of course, is that for series the right-hand-side could be infinite even if the left-hand-side is finite.

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