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Canonical static concept record

Inner-product space

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Summary

A vector space with a positive-definite inner product.

Record metadata

Canonical name: Inner-product space
Concept ID: inner_product_space
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Analysis
Domains: Analysis, Algebra
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

real or complex vector space $V$

Data

inner product $\langle \cdot,\cdot \rangle$

Axioms / constraints

linearity/sesquilinearity
conjugate symmetry
positive definiteness

Canonically induces

norm $\lVert x\rVert=\sqrt{\langle x,x\rangle}$
metric and topology

Notes

The inner product canonically induces a norm. The complex convention is Hermitian sesquilinear.

Concept 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
Princeton — Functional Analysis Lecture Notes — Functional Analysis Lecture Notes · course notes · source ID princeton-functional-analysis

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Vector space Inner-product space

Permalink to relation

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

Relation ID: vector_space_to_inner_product
Relation type: Add data add-data
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: + inner product

Authored explanation

For a real or complex vector space, choose a positive-definite symmetric bilinear form (real) or Hermitian sesquilinear form (complex).

How to interpret this relation type

Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.

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
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.

Inner-product space Hilbert space

Permalink to relation

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

Relation ID: e_inner_product_hilbert
Relation type: Impose axiom impose-axiom
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: require completeness

Authored explanation

Require completeness for the norm induced by the inner product.

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 — Hilbert space — Hilbert space · expository · source ID wp-hilbert
nLab — Hilbert space — Hilbert space · reference · source ID nlab-hilbert
Princeton — Functional Analysis Lecture Notes — Functional Analysis Lecture Notes · course notes · source ID princeton-functional-analysis

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

Inner-product space Quantum expectation value

Permalink to relation

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

Relation ID: innerproduct_to_expectation
Relation type: Mathematical formulation mathematical-formulation
Direction: source → target
Endpoint roles: source: Mathematically formulates; target: Mathematically formulated using
Authored annotation: pure-state quadratic form

Authored explanation

For a normalized state vector $\psi$ in the domain of $A$ , the pure-state expectation is $\langle\psi,A\psi\rangle$ ; for an unnormalized nonzero vector, divide by $\langle\psi,\psi\rangle$ .

How to interpret this relation type

A mathematical concept supplies part of the formal language, state space, representation, or analytic machinery used by a scientific or mathematical-physics concept. The source is the mathematical predecessor; this does not claim that physical content follows from mathematics alone.

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
Cambridge — Quantum Mechanics — Quantum Mechanics · University of Cambridge lecture notes · source ID tong-quantum

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Inner-product space

Summary

Record metadata

Carrier(s)

Data

Axioms / constraints

Canonically induces

Notes

Concept sources

Incoming relations (arrows to this concept)

Vector space Inner-product space

Authored explanation

How to interpret this relation type

Relation sources

Outgoing relations (arrows from this concept)

Inner-product space Hilbert space

Authored explanation

How to interpret this relation type

Relation sources

Inner-product space Normed vector space

Authored explanation

How to interpret this relation type

Relation sources

Inner-product space Quantum expectation value

Authored explanation

How to interpret this relation type

Relation sources