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Let $(X,\|\cdot\|)$ be a normed space. Is it true that $\inf\{\|x+y\|:x,y\in X\}\le\inf\{\|x\| +\|y\|:x,y\in X\}$?

I need it in order to finish a proof. It seems easy, but I don't see the proof. Could anyone please help?

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  • $\begingroup$ What is a closed normed space? $\endgroup$ Nov 3, 2021 at 19:43
  • $\begingroup$ Seems to follow straight from triangle inequality. $\endgroup$ Nov 3, 2021 at 19:50
  • $\begingroup$ @JoséCarlosSantos oops, my bad! I added more than I needed. I'll change it. It had to do with a closed subspace of a normed space, but I didn't remove the 'closed' for whatever reason. $\endgroup$ Nov 3, 2021 at 20:20
  • $\begingroup$ If $X$ is a normed space, then both sides are $0$, so the inequality ${}\le {}$ is true $\endgroup$
    – GEdgar
    Nov 3, 2021 at 20:23
  • $\begingroup$ @PotemkinMetroCard Really? How can it be so direct? $\endgroup$ Nov 3, 2021 at 20:24

1 Answer 1

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Let $\nu=\inf \{\|x+y\|:x,y\in X\}$ and pick $x,y\in X$ arbitrary. Then $$\nu\leq \|x+y\|\leq \|x\|+\|y\|$$ This shows $\nu$ is a lower bound of $\{\|x\|+\|y\|:x,y\in X\}$. Therefore $$\nu \leq \inf \{\|x\|+\|y\|:x,y\in X\}$$ and we're done.

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  • $\begingroup$ Oh, I see. Thank you for answering! $\endgroup$ Nov 3, 2021 at 20:28

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