LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-01

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The quotient seminorm satisfies the triangle inequality

Statement

Let $X$ be a normed space and let $M \leq X$ . Then for all $x, y \in X$ ,

$∥ (x + y) + M ∥_{X / M} \leq ∥ x + M ∥_{X / M} + ∥ y + M ∥_{X / M} .$

Facts & Assumptions

Given: A normed space $X$ , a linear subspace $M \leq X$ , vectors $x, y \in X$ , and a real $ε > 0$ .

[L1]

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

[L2]

The quotient seminorm is representative-independent, so $(x + y) + (m_{1} + m_{2}) + M = (x + y) + M$ may be read with any $m_{1}, m_{2} \in M$ (The quotient seminorm is independent of the chosen coset representative).

Proof

technique · direct

1.1

By [L1], choose $m_{1}, m_{2} \in M$ such that $∥ x + m_{1} ∥ < ∥ x + M ∥_{X / M} + ε / 2$ and $∥ y + m_{2} ∥ < ∥ y + M ∥_{X / M} + ε / 2$ .

L1choose

2.1

Since $m_{1} + m_{2} \in M$ , [L2] lets us evaluate the quotient seminorm of $(x + y) + M$ at the representative $x + y + m_{1} + m_{2}$ . Hence $∥ (x + y) + M ∥_{X / M} \leq ∥ x + y + m_{1} + m_{2} ∥ \leq ∥ x + m_{1} ∥ + ∥ y + m_{2} ∥ < ∥ x + M ∥_{X / M} + ∥ y + M ∥_{X / M} + ε$ by step 1.1.

step 1.1L1L2algebra

3.1

Since $ε > 0$ was arbitrary, the displayed strict inequality of step 2.1 implies the stated triangle inequality.

step 2.1given∎

Depends on

Used by

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

Dependency tree · two levels

5 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)
Theo Buhler and Dietmar A. Salamon, Functional Analysis (standard reference, not scraped)