functional
Let an agent explain functionals and their kinds, relay their role in functional analysis, the calculus of variations and physics from mathematics references, describe the constructions the registry aliases name, and distinguish functionals from functions, operators and higher-order functions in programming.
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 functionals and their kinds, relay their role in functional analysis, the calculus of variations and physics from mathematics references, describe the constructions the registry aliases name, and distinguish functionals from functions, operators and higher-order functions in programming.
In mathematics, a map from a space of functions or vectors to its field of scalars, as in linear functionals of functional analysis, integral functionals such as the action and energy functionals of physics that are minimised in the calculus of variations, and the modulus of continuity as a functional measuring uniform continuity; related constructions include the blossom, a symmetric multi-affine map associated with a polynomial in spline theory, and entanglement witnesses, functionals used to detect quantum entanglement.
What it is for: Not applicable; a mathematical object.
It can be explain the concept; relay roles in analysis and physics; describe named constructions; distinguish related concepts.
Distinguishing features
Scalar output
Function inputs
Linear and nonlinear
Variational use
What it looks like
Not a visible object; a mapping.
Physical character
Riesz representation theorem: 1907-1909 note - linear functionals on Hilbert spaces
calculus of variations: 17th-18th centuries note - Euler, Lagrange
dual space: set of continuous linear functionals note
How it is recognised
Map from functions or vectors to scalars
Integral functional, energy functional, modulus of continuity, blossom, entanglement witness
Functions map numbers; operators map functions to functions; higher-order functions are programming constructs
Related models
is a kind of - in registry terms
is studied in -
is minimised in -
is contrasted with - which returns functions
In practice
Families and kinds
linear functionals and dual spaces
integral functionals
energy and action functionals
functionals measuring regularity such as the modulus of continuity
blossoms in spline theory
entanglement witnesses in quantum information
Standards and regulation
No regulation
Failure modes and hazards
Confusing functionals with operators
Registry aliases naming loosely related constructions
Assuming linearity
Also called
Where this came from
wikidata · CC0 1.0
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 functional is.
Science.
Definition
Definition.
Definition
Definition.
- What is a functional, and how does it differ from a function, operator and programming higher-order function? definition
- Is the question about analysis, physics, splines or quantum information? boundary
Examples
Named constructions.
Examples
Examples.
- What are integral and energy functionals, the modulus of continuity, blossoms and entanglement witnesses? definition
- Which entry fits the specific construction? action
Analysis Functional analysis.
Science.
Linear
Linear functionals.
Linear
Linear.
- What are linear functionals, dual spaces and the Riesz and Hahn-Banach theorems? provenance
- Which references are standard? provenance
Variations
Calculus of variations.
Variations
Variations.
- How are functionals minimised with Euler-Lagrange equations? provenance
- Which sources are cited? provenance
Applications Applications.
Application.
Physics
Physics.
Physics
Physics.
- How do action and energy functionals and density functional theory use functionals? provenance
- Which entry fits density functional theory? action
Quantum
Quantum information.
Quantum
Quantum.
- How do entanglement witnesses detect entanglement? provenance
- Which entry fits entanglement witness? action
Context Splines and history.
Context.
Splines
Blossoms.
Splines
Splines.
- What is the blossoming principle in spline theory? provenance
- Which entry fits blossom (functional)? action
History
History.
History
History.
- How did Volterra, Hadamard and others develop functionals? provenance
- Which entry fits the history of functional analysis? action
What the second pass must settle
- Should blossom and entanglement witness be separate primary entries?
- How should mathematics references be linked?
- The registry entry has merged aliases naming constructions from different fields; should they be split off?