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In an comment under an answer to this question

How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$?

it is claimed that we have the inequality $\|x\| \le\|[x]\| +\epsilon$, which relates the norm of a space $X$ to the quotient norm on the quotient space $X/Y$. We assume that $X$ is complete and $Y$ is a closed subspace.

I can see that in many cases (such as the proof in the linked question) we can assume that this inequality holds w.l.o.g., but I don't see how it can hold as such.

Is the inequality $\|x\| \le\|[x]\| +\epsilon$ true and if so how can we prove it?

Hanno
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jcutler
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1 Answers1

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The notation in the other post (as well as in your question here) is a little misleading and is likely the source of your confusion.

Let $x \in X$, and let $[x] = x + Y$ be the equivalence class of $x$ in $X/Y$. The claim is

Let $\epsilon > 0$. There exists $\tilde{x} \in [x]$ (possibly different from $x$) such that $\|\tilde{x}\| < \|[x]\| + \epsilon$.

Note that it is not true that $\|x\| < \|[x]\| + \epsilon$ for any $x \in X$.

To prove the claim, use the definition of the norm on $X/Y$. Since $\|[x]\| := \inf_{y \in Y} \|x + y\|$, there exists some $y \in Y$ such that $\|x+y\| < \|[x]\| + \epsilon$. Let $\tilde{x} := x+y$.

angryavian
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