Crate amari_functional 

Functional analysis on multivector spaces.

This crate provides the mathematical foundations of functional analysis applied to Clifford algebra-valued function spaces, including:

• Hilbert spaces: Complete inner product spaces over multivectors
• Linear operators: Bounded and unbounded operators on Hilbert spaces
• Spectral theory: Eigenvalues, eigenvectors, and spectral decomposition
• Compact operators: Fredholm theory and index theorems
• Sobolev spaces: Function spaces with weak derivatives

Space Hierarchy

The module implements the standard functional analysis hierarchy:

VectorSpace
    ↓
NormedSpace (adds ||·||)
    ↓
BanachSpace (complete normed space)
    ↓
InnerProductSpace (adds ⟨·,·⟩)
    ↓
HilbertSpace (complete inner product space)

Quick Start

Finite-Dimensional Hilbert Space

use amari_functional::space::{MultivectorHilbertSpace, HilbertSpace, InnerProductSpace};

// Create the Hilbert space Cl(2,0,0) ≅ ℝ⁴
let space: MultivectorHilbertSpace<2, 0, 0> = MultivectorHilbertSpace::new();

// Create elements
let x = space.from_coefficients(&[1.0, 2.0, 0.0, 0.0]).unwrap();
let y = space.from_coefficients(&[0.0, 0.0, 3.0, 4.0]).unwrap();

// Compute inner product
let ip = space.inner_product(&x, &y);
assert!(ip.abs() < 1e-10); // x and y are orthogonal

L² Space of Multivector Functions

use amari_functional::space::{MultivectorL2, L2Function, NormedSpace};
use amari_core::Multivector;

// Create L²([0,1], Cl(2,0,0))
let l2: MultivectorL2<2, 0, 0> = MultivectorL2::unit_interval();

// Define a function f(x) = x * e₀ (scalar coefficient is x)
let f = L2Function::new(|x| {
    Multivector::<2, 0, 0>::scalar(x[0])
});

// Compute L² norm: ||f||² = ∫₀¹ x² dx = 1/3
let norm_sq = l2.norm(&f).powi(2);
assert!((norm_sq - 1.0/3.0).abs() < 0.1);

Mathematical Background

This crate is built on the following mathematical concepts:

• Clifford algebras Cl(P,Q,R): 2^(P+Q+R)-dimensional real algebras
• Hilbert spaces: Complete inner product spaces with orthogonal decomposition
• Bounded operators: Continuous linear maps between Hilbert spaces
• Spectral theorem: Self-adjoint operators have orthonormal eigenbases
• Compact operators: Operators mapping bounded sets to precompact sets

Features

• std (default): Enable standard library features
• parallel: Enable Rayon parallelism for large-scale computations
• formal-verification: Enable Creusot contracts for formal verification

Re-exports

pub use space::BanachSpace;

pub use space::Domain;

pub use space::HilbertSpace;

pub use space::InnerProductSpace;

pub use space::L2Function;

pub use space::MultivectorHilbertSpace;

pub use space::MultivectorL2;

pub use space::NormedSpace;

pub use space::VectorSpace;

pub use operator::AdjointableOperator;

pub use operator::BoundedOperator;

pub use operator::CompactMatrixOperator;

pub use operator::CompactOperator;

pub use operator::CompositeOperator;

pub use operator::FiniteRankOperator;

pub use operator::FredholmMatrixOperator;

pub use operator::FredholmOperator;

pub use operator::IdentityOperator;

pub use operator::LinearOperator;

pub use operator::MatrixOperator;

pub use operator::OperatorNorm;

pub use operator::ProjectionOperator;

pub use operator::ScalingOperator;

pub use operator::SelfAdjointOperator;

pub use operator::ZeroOperator;

pub use spectral::compute_eigenvalues;

pub use spectral::inverse_iteration;

pub use spectral::power_method;

pub use spectral::spectral_decompose;

pub use spectral::Eigenpair;

pub use spectral::Eigenvalue;

pub use spectral::SpectralDecomposition;

pub use sobolev::poincare_constant_estimate;

pub use sobolev::H1Space;

pub use sobolev::H2Space;

pub use sobolev::SobolevFunction;

pub use sobolev::SobolevSpace;

Modules

operator

Linear operators on function spaces.

sobolev

Sobolev spaces for multivector-valued functions.

space

Function space abstractions for functional analysis.

spectral

Spectral theory for linear operators.

Structs

Bounded

Bounded operator: ||Tx|| ≤ M||x|| for some M and all x.

Compact

Compact operator: maps bounded sets to precompact sets.

Complete

Complete space: every Cauchy sequence converges within the space.

ContinuousSpectrum

Continuous spectrum: contains intervals or continuous parts.

DiscreteSpectrum

Discrete spectrum: consists only of isolated eigenvalues.

Fredholm

Fredholm operator: finite-dimensional kernel and cokernel.

General

General operator with no special symmetry properties.

H1Regularity

H¹ regularity (k=1 Sobolev space).

H2Regularity

H² regularity (k=2 Sobolev space).

HkRegularity

General Hᵏ regularity.

L2Regularity

L² regularity (k=0 Sobolev space).

MixedSpectrum

Mixed spectrum: combination of discrete and continuous parts.

Multivector

A multivector in a Clifford algebra Cl(P,Q,R)

NonCompact

Non-compact operator.

Normal

Normal operator: TT* = T*T.

NotFredholm

Not Fredholm.

PreHilbert

Pre-Hilbert space: inner product defined but not necessarily complete.

Properties

Zero-cost wrapper for phantom type properties.

PurePointSpectrum

Pure point spectrum: all spectral values are eigenvalues.

Scalar

Scalar type - wrapper around Multivector with only grade 0

SelfAdjoint

Self-adjoint operator: T = T*.

SemiFredholm

Semi-Fredholm operator: either kernel or cokernel is finite-dimensional.

Unbounded

Unbounded operator: no finite bound exists.

Unitary

Unitary operator: TT = TT = I.

Enums

FunctionalError

Errors that can occur during functional analysis operations.

Traits

BoundednessProperty

Marker trait for operator boundedness properties.

CompactnessProperty

Marker trait for operator compactness properties.

CompletenessProperty

Marker trait for space completeness properties.

FredholmProperty

Marker trait for Fredholm operator properties.

RegularityProperty

Marker trait for Sobolev regularity.

SpectralProperty

Marker trait for spectral properties.

SymmetryProperty

Marker trait for operator symmetry properties.

Type Aliases

CompactSelfAdjointOperator

Properties of a compact self-adjoint operator.

H1SpaceProperties

Standard H¹ Sobolev space properties.

HilbertSchmidtOperator

Properties of a Hilbert-Schmidt operator.

L2SpaceProperties

Standard L² Hilbert space properties.

Result

Result type for functional analysis operations.

UnitaryOperator

Properties of a unitary operator.