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The quotient seminorm is a norm exactly when the subspace is closed

Statement

Let $X$ be a normed space and let $M \leq X$ . The quotient seminorm on $X / M$ is a norm if and only if $M$ is closed in $X$ .

Facts & Assumptions

Given: A normed space $X$ , a linear subspace $M \leq X$ , and a vector $x \in X$ .

[L1]

The quotient seminorm is $∥ x + M ∥_{X / M} = dist (x, M)$ (The quotient seminorm (|x+M|{X/M}=\inf{m\in M}|x+m|=\operatorname{dist}(x,M))).

[L2]

A linear subspace contains $0$ , so $M$ is nonempty (Linear subspace of a vector space).

[L3]

For a nonempty subset $A$ of a metric space, the closure of $A$ is exactly ${u : dist (u, A) = 0}$ , and a set is closed exactly when it equals its closure (The closure of a nonempty $A$ is ${x : d (x, A) = 0}$ , equals $A$ together with its limit points, and is the smallest closed superset).

Proof

technique · direct

1.1

By [L2] and [L3], $dist (x, M) = 0$ exactly when $x \in \overline{M}$ . Therefore [L1] gives $∥ x + M ∥_{X / M} = 0$ exactly when $x \in \overline{M}$ .

L1L2L3

2.1

If $M$ is closed and $∥ x + M ∥_{X / M} = 0$ , then step 1.1 gives $x \in \overline{M} = M$ . Hence $x + M = M$ , the zero coset. So the quotient seminorm is definite and therefore a norm.

step 1.1L3

2.2

Conversely, assume the quotient seminorm is a norm. If $M$ were not closed, then [L3] would give some $x \in \overline{M} ∖ M$ . Step 1.1 would then give $∥ x + M ∥_{X / M} = 0$ , while $x + M \neq M$ because $x \notin M$ , contradicting definiteness. Therefore $M$ is closed.

step 1.1L3assume-contradischarge-contradiction

3.1

Steps 2.1 and 2.2 prove the equivalence.

step 2.1step 2.2∎

Depends on

Used by

Dependency tree · two levels

24 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Gerald Teschl, Topics in Real and Functional Analysis (standard reference, not scraped)
Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis (standard reference, not scraped)