# Is it true that $\inf\{\|x+y\|:x\in X, y\in Y\}\le\inf\{\|x\| +\|y\|:x,y\in X\}$?

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?

• What is a closed normed space? Nov 3, 2021 at 19:43
• Seems to follow straight from triangle inequality. Nov 3, 2021 at 19:50
• @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. Nov 3, 2021 at 20:20
• If $X$ is a normed space, then both sides are $0$, so the inequality ${}\le {}$ is true Nov 3, 2021 at 20:23
• @PotemkinMetroCard Really? How can it be so direct? Nov 3, 2021 at 20:24

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.