LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The quotient seminorm is independent of the chosen coset representative

Statement

Let $X$ be a normed space and let $M \leq X$ . If $x + M = x^{'} + M$ in $X / M$ , then

$\inf_{m \in M} ∥ x + m ∥ = \inf_{m \in M} ∥ x^{'} + m ∥ .$

Therefore the quotient seminorm of The quotient seminorm (|x+M|{X/M}=\inf{m\in M}|x+m|=\operatorname{dist}(x,M)) is well defined.

Facts & Assumptions

Given: A normed space $X$ , a linear subspace $M \leq X$ , and representatives $x, x^{'} \in X$ with $x + M = x^{'} + M$ .

[L1]

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

[L2]

Two cosets are equal exactly when their representatives differ by an element of $M$ (Coset equality, well-defined quotient operations, and the canonical projection with kernel $W$ ).

Proof

technique · direct

1.1

By [L2], there is $m_{0} \in M$ with $x^{'} = x + m_{0}$ . For every $m \in M$ , $x^{'} + m = x + (m_{0} + m)$ , and $m_{0} + m$ still lies in $M$ . Thus ${∥ x^{'} + m ∥ : m \in M} \subseteq {∥ x + n ∥ : n \in M}$ .

2.1

The same argument with the roles of $x$ and $x^{'}$ reversed gives the reverse inclusion, so the two sets of admissible norms are equal. Their infima are therefore equal, which is exactly the claim of [L1].

step 1.1L1L2∎

Depends on

Used by

Cited to discharge well-definedness by The quotient seminorm (‖x+M‖_X/M=inf_m∈ M‖x+m‖=dist(x,M)).

Dependency tree · two levels

7 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)