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Research draft

norm

vr.tr.norm · XCT.QLT

Let an agent explain norms in mathematics, relay axioms, examples, induced metrics and applications from mathematics sources, describe the named norms and flag misfiled aliases, and distinguish norms from metrics, inner products, seminorms and social or technical norms.

Thing Registry Cross-cutting context

Research draft, second pass

A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.

written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify

Researched by: Claude

Purpose and description

Let an agent explain norms in mathematics, relay axioms, examples, induced metrics and applications from mathematics sources, describe the named norms and flag misfiled aliases, and distinguish norms from metrics, inner products, seminorms and social or technical norms.

A function assigning a non-negative length or size to each vector in a vector space, satisfying positive definiteness, absolute homogeneity and the triangle inequality, including the Euclidean norm, the canonical norm induced by an inner product, p-norms such as the taxicab norm for p equal 1 and the maximum norm, and specialised norms such as the Bombieri norm on polynomials; a seminorm relaxes positive definiteness. The registry aliases edit distance, a metric on strings, and Deficiency, a statistics term, are not norms and appear misfiled. The word norm also names social norms and standards, which are different senses.

What it is for: Measuring size of vectors and functions.

It can be explain axioms; relay examples; describe named norms; flag misfiled aliases.

Distinguishing features

Non-negative size

Triangle inequality

Induces a metric

Many examples

What it looks like

Not a physical object; written with double vertical bars.

Physical character

axioms: 3 count - positivity, homogeneity, triangle inequality

Euclidean norm: square root of sum of squares note

Banach space: complete normed space note - Banach, 1920s

How it is recognised

Length function on a vector space

Deficiency (statistics), canonical norm induced by inner product, Euclidean norm, p-norm, Bombieri norm, edit distance

Metrics measure distance between points; inner products give angles; seminorms allow zero length for nonzero vectors; social norms are behavioural

Related models

is a kind of - in registry terms

seminorm

induces -

metric (mathematics)

includes -

Euclidean norm

is contrasted with - misfiled alias

edit distance

In practice

Families and kinds

p-norms

inner product norms

operator and matrix norms

function space norms

polynomial norms

Standards and regulation

No regulation

Failure modes and hazards

Confusing norms and metrics

Misfiled string and statistics aliases

Conflating mathematical and social norms

Also called

Deficiency (statistics)canonical norm induced by inner productEuclidean normp-normBombieri normedit distanceSchatten normmatrix normoperator normnatural norm

Where this came from

wikidata · CC0 1.0

Source repository Registry entry point

Drafted structure

Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.

Understand What a norm is.

Definition.

Definition

Definition.

Definition

Definition.

  1. What is a norm, and how does it differ from metrics, inner products, seminorms and social norms? definition
  2. Is the question about linear algebra, analysis, machine learning or another sense of norm? boundary

Named norms

Named norms.

Named norms

Named norms.

  1. What are Euclidean, p-norms, inner product and Bombieri norms, and why are edit distance and Deficiency different? definition
  2. Which entry fits the specific norm? action
Theory Theory.

Science.

Axioms

Axioms.

Axioms

Axioms.

  1. What do the norm axioms require, and how do seminorms differ? provenance
  2. Which references are standard? provenance

Banach spaces

Normed spaces.

Banach spaces

Banach spaces.

  1. What are normed and Banach spaces? provenance
  2. Which sources are cited? provenance
Applications Applications.

Science.

Machine learning

Regularisation.

Machine learning

Machine learning.

  1. How are L1 and L2 norms used in regularisation? provenance
  2. Which entry fits regularization (mathematics)? action

Numerical

Error measures.

Numerical

Numerical.

  1. How are norms used to measure errors in numerical analysis? measurement
  2. Which entry fits matrix norm? action
Context Other senses.

Context.

Strings

Edit distance.

Strings

Strings.

  1. What is edit distance, and why is it a metric rather than a norm? provenance
  2. Which entry fits Levenshtein distance? action

Social norms

Social norms.

Social norms

Social norms.

  1. How does the mathematical sense differ from social norms? provenance
  2. Which entry fits social norm? action

What the second pass must settle

  • The registry aliases edit distance and Deficiency (statistics) should be moved to their own entries
  • Should matrix norms be a separate entry?
  • How should functional analysis references be linked?
Draft as JSON