# Does $\|\sum_{i=1}^{\infty}x_i\| \leq \sum_{i=1}^{\infty}\|x_i\|$ hold for a norm?

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..:(

• Definitely holds for finite sums - try induction on $n$. – stochasticboy321 Dec 5 '15 at 19:01
• How do you perform an uncountable sum? Which order are you taking for computing the sum – mathcounterexamples.net Dec 5 '15 at 19:02
• @stochasticboy321 Sorry, a typo. Changed it. I guess It's true for infinite summation because $||x||$ is a continuous function? – 826313315 Dec 5 '15 at 21:16