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Canonical static concept record

Normed vector space

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Summary

A seminormed vector space whose seminorm separates points.

Record metadata

Canonical name: Normed vector space
Concept ID: normed_vector_space
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Analysis
Domains: Analysis, Topology, Algebra
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

seminormed vector space $V$ over a field with a specified real-valued absolute value

Data

norm $\lVert \cdot \rVert$

Axioms / constraints

seminorm axioms
$\lVert x \rVert =0 \Rightarrow x=0$

Canonically induces

Metric $d(x,y)=\lVert x-y \rVert$
topological vector space

Notes

Its norm canonically induces a metric, uniformity, topology, and topological-vector-space structure.

Concept sources

Wikipedia — Normed vector space — Normed vector space · expository · source ID wp-normed
Mathlib — Seminorms — Seminorms · formal reference · source ID mathlib-seminorm
Mathlib — Normed modules — Normed modules · formal reference · source ID mathlib-normed-module

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Inner-product space Normed vector space

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: inner_product_to_normed
Relation type: Induced / forgotten induced-forgotten
Direction: source → target
Endpoint roles: source: Yields by induction / forgetting; target: Obtained by induction / forgetting from
Authored annotation: induce $\lVert x\rVert=\sqrt{\langle x,x\rangle}$

Authored explanation

Retain the canonical norm induced by the inner product.

How to interpret this relation type

Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.

Relation sources

Wikipedia — Inner product space — Inner product space · expository · source ID wp-inner-product
nLab — Inner product space — Inner product space · reference · source ID nlab-inner-product-space
Wikipedia — Normed vector space — Normed vector space · expository · source ID wp-normed
Princeton — Functional Analysis Lecture Notes — Functional Analysis Lecture Notes · course notes · source ID princeton-functional-analysis

Seminormed vector space Normed vector space

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: e_seminorm_norm
Relation type: Impose axiom impose-axiom
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: + definiteness

Authored explanation

Require $p(x)=0$ only for $x=0$ .

How to interpret this relation type

Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.

Relation sources

Wikipedia — Normed vector space — Normed vector space · expository · source ID wp-normed
nLab — Norm — Norm · reference · source ID nlab-norm
Princeton — Functional Analysis Lecture Notes — Functional Analysis Lecture Notes · course notes · source ID princeton-functional-analysis

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.