Hilbert Space Ontology

flow "HilbertSpaceOntology" v1.0

# ============================================================
# HILBERT SPACE ONTOLOGY — FLOW ALGEBRA
# ============================================================

SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/functional-analysis/hilbert-space-ontology.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Functional Analysis"
ENTITY := "Hilbert Space"

# ---------- CORE ENTITY ----------
HILBERT_SPACE := {
    type: CompleteInnerProductSpace,
    domain: FunctionalAnalysis,
    purpose: "Represent an inner-product vector space that is complete in the norm induced by its inner product, supporting orthogonality, projection, bases and spectral methods."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    VECTOR_SPACE,
    INNER_PRODUCT,
    NORM,
    METRIC,
    COMPLETENESS,
    HILBERT_SPACE,
    BANACH_SPACE,
    ORTHOGONALITY,
    ORTHONORMAL_SET,
    ORTHONORMAL_BASIS,
    PROJECTION,
    CLOSED_SUBSPACE,
    RIESZ_REPRESENTATION,
    ADJOINT_OPERATOR,
    SELF_ADJOINT_OPERATOR
}

# ---------- BASIC STRUCTURE ----------
HILBERT_SPACE
    := VECTOR_SPACE
       + INNER_PRODUCT
       + COMPLETENESS

INNER_PRODUCT
    -> induces(NORM)

NORM
    -> induces(METRIC)

COMPLETENESS
    -> measured_with(INDUCED_NORM)

# ---------- INNER PRODUCT ----------
INNER_PRODUCT
    := <x,y>

INNER_PRODUCT_PROPERTIES := {
    LINEARITY_OR_SESQUILINEARITY,
    CONJUGATE_SYMMETRY,
    POSITIVE_DEFINITENESS
}

for complex Hilbert spaces:
    convention
        -> must_specify(which_argument_is_linear)

# ---------- INDUCED NORM ----------
||x||
    := sqrt(<x,x>)

INNER_PRODUCT_NORM
    -> satisfies(PARALLELOGRAM_IDENTITY)

PARALLELOGRAM_IDENTITY:

||x+y||^2
+
||x-y||^2
=
2||x||^2
+
2||y||^2

# ---------- METRIC ----------
d(x,y)
    := ||x-y||

HILBERT_SPACE
    -> COMPLETE_METRIC_SPACE
       under(INDUCED_METRIC)

# ---------- COMPLETENESS ----------
sequence
    := (x_n)

if
    (x_n) is Cauchy
then

exists x in H
such_that

||x_n-x||
    -> 0

COMPLETENESS
    -> distinguishes(HILBERT_SPACE_FROM_INCOMPLETE_INNER_PRODUCT_SPACE)

# ---------- BANACH RELATION ----------
HILBERT_SPACE
    -> BANACH_SPACE
       under(INNER_PRODUCT_NORM)

BANACH_SPACE
    -/> necessarily(HILBERT_SPACE)

NORMED_SPACE
    -> Hilbertisable_only_if(NORM_ARISES_FROM_INNER_PRODUCT)

# ---------- CAUCHY-SCHWARZ ----------
abs(<x,y>)
    <= ||x||*||y||

CAUCHY_SCHWARZ_INEQUALITY
    -> fundamental_to(HILBERT_GEOMETRY)

# ---------- ORTHOGONALITY ----------
x
    -> orthogonal_to(y)

iff

<x,y>
    = 0

ORTHOGONALITY
    -> symmetric_under(INNER_PRODUCT_CONVENTIONS)

ORTHOGONAL_VECTORS
    -> satisfy(PYTHAGOREAN_IDENTITY)

if
    x orthogonal_to y
then

||x+y||^2
    =
||x||^2 + ||y||^2

# ---------- ORTHONORMAL SET ----------
ORTHONORMAL_SET
    := {
        e_i :
        <e_i,e_j> = delta_ij
    }

ORTHONORMAL
    -> ORTHOGONAL
    -> UNIT_NORM

# ---------- ORTHONORMAL BASIS ----------
ORTHONORMAL_BASIS
    -> orthonormal_set
    -> complete_span_dense_in(H)

for suitable indexing:

x
    = sum(
        <x,e_n>*e_n
    )

with convergence_in(HILBERT_NORM)

ORTHONORMAL_BASIS
    -> enables(FOURIER_EXPANSION)

# ---------- BESSEL INEQUALITY ----------
for orthonormal sequence (e_n):

sum(
    abs(<x,e_n>)^2
)
    <= ||x||^2

BESSEL_INEQUALITY
    -> bounds(FOURIER_COEFFICIENT_ENERGY)

# ---------- PARSEVAL IDENTITY ----------
if
    (e_n)
        -> complete_orthonormal_basis
then

||x||^2
    =
sum(
    abs(<x,e_n>)^2
)

PARSEVAL_IDENTITY
    -> expresses(NORM_VIA_BASIS_COEFFICIENTS)

# ---------- CLOSED SUBSPACE ----------
M
    -> CLOSED_LINEAR_SUBSPACE_OF(H)

then

M
    -> HILBERT_SPACE
       under(RESTRICTED_INNER_PRODUCT)

CLOSEDNESS
    -> preserves(COMPLETENESS)

# ---------- ORTHOGONAL COMPLEMENT ----------
M_perp
    := {
        x in H :
        <x,m> = 0
        for_all m in M
    }

M_perp
    -> CLOSED_SUBSPACE

if
    M closed
then

H
    = M direct_sum M_perp

# ---------- PROJECTION THEOREM ----------
for
    closed_subspace M
    and x in H

exists unique
    m in M

such_that

||x-m||
    = inf(
        ||x-y|| :
        y in M
    )

m
    := P_M(x)

P_M
    -> ORTHOGONAL_PROJECTION

# ---------- PROJECTION PROPERTIES ----------
P_M^2
    = P_M

P_M
    -> bounded_linear_operator

P_M
    -> self_adjoint

RANGE(P_M)
    = M

KERNEL(P_M)
    = M_perp

# ---------- RIESZ REPRESENTATION ----------
for every
    continuous_linear_functional f on H

exists unique
    y in H

such_that

f(x)
    = <x,y>

subject_to
    chosen_inner_product_convention

RIESZ_REPRESENTATION_THEOREM
    -> identifies(H_WITH_DUAL_UP_TO_CONJUGATION_CONVENTION)

# ---------- ADJOINT OPERATOR ----------
for bounded operator
    T: H -> H

ADJOINT
    := T_star

such_that

<Tx,y>
    =
<x,T_star y>

for_all x,y

ADJOINT_OPERATOR
    -> uniquely_determined

# ---------- SELF-ADJOINT OPERATOR ----------
T
    -> SELF_ADJOINT

iff

T
    = T_star

SELF_ADJOINT_OPERATOR
    -> has(REAL_SPECTRUM)

SELF_ADJOINT_OPERATOR
    -> central_to(SPECTRAL_THEORY)

# ---------- UNITARY OPERATOR ----------
U
    -> UNITARY

iff

U_star U
    = I
    AND
U U_star
    = I

UNITARY_OPERATOR
    -> preserves(INNER_PRODUCT)
    -> preserves(NORM)

# ---------- NORMAL OPERATOR ----------
T
    -> NORMAL

iff

T*T_star
    =
T_star*T

SELF_ADJOINT
    -> NORMAL

UNITARY
    -> NORMAL

NORMAL_OPERATOR
    -> admits(SPECTRAL_THEOREM_FOR_APPROPRIATE_CONTEXT)

# ---------- COMPACT OPERATORS ----------
COMPACT_OPERATOR
    -> maps(BOUNDED_SETS_TO_RELATIVELY_COMPACT_SETS)

COMPACT_SELF_ADJOINT_OPERATOR
    -> has(SPECTRAL_STRUCTURE_ANALOGOUS_TO_SYMMETRIC_MATRICES)

NONZERO_EIGENVALUES
    -> discrete
       with(FINITE_MULTIPLICITY)
       under(COMPACT_SELF_ADJOINT_CONTEXT)

EIGENVECTORS
    -> may_form(ORTHONORMAL_BASIS_FOR_RELEVANT_SUBSPACE)

# ---------- L2 SPACE ----------
L2(Omega)
    -> HILBERT_SPACE

INNER_PRODUCT:

<f,g>
    :=
integral_Omega(
    f(x)*conjugate(g(x))
    dx
)

NORM:
    ||f||_2
        = sqrt(<f,f>)

L2
    -> fundamental_function_space

# ---------- l2 SEQUENCE SPACE ----------
l2
    := {
        x=(x_n) :
        sum(abs(x_n)^2) < infinity
    }

INNER_PRODUCT:
    <x,y>
        = sum(
            x_n*conjugate(y_n)
          )

l2
    -> HILBERT_SPACE

# ---------- FOURIER ANALYSIS ----------
ORTHONORMAL_FUNCTION_SYSTEM
    -> provides(FOURIER_COEFFICIENTS)

c_n
    := <f,e_n>

f
    -> represented_by(ORTHONORMAL_EXPANSION)

HILBERT_SPACE
    -> provides(GEOMETRIC_FRAMEWORK_FOR_FOURIER_ANALYSIS)

# ---------- WEAK CONVERGENCE ----------
x_n
    -> weakly_converges_to(x)

iff

<x_n,y>
    -> <x,y>

for_all
    y in H

RIESZ_REPRESENTATION
    -> converts(DUAL_TESTING_TO_INNER_PRODUCT_TESTING)

NORM_CONVERGENCE
    -> implies(WEAK_CONVERGENCE)

# ---------- SEPARABILITY ----------
SEPARABLE_HILBERT_SPACE
    -> has(COUNTABLE_DENSE_SUBSET)

in infinite-dimensional separable Hilbert space:
    -> admits(COUNTABLE_ORTHONORMAL_BASIS)

all infinite-dimensional separable Hilbert spaces
    -> isometrically_isomorphic_to(l2)
       over(SAME_SCALAR_FIELD)

# ---------- APPLICATIONS ----------
HILBERT_SPACE
    -> applied_to(QUANTUM_MECHANICS)
    -> applied_to(FOURIER_ANALYSIS)
    -> applied_to(PDEs)
    -> applied_to(SIGNAL_PROCESSING)
    -> applied_to(OPTIMISATION)
    -> applied_to(STATISTICS)
    -> applied_to(SPECTRAL_THEORY)

# ---------- RELATION GRAPH ----------
VECTOR_SPACE
    + INNER_PRODUCT
    -> INNER_PRODUCT_SPACE

INNER_PRODUCT
    -> induces(NORM)

INNER_PRODUCT_SPACE
    + COMPLETENESS
    -> HILBERT_SPACE

HILBERT_SPACE
    -> BANACH_SPACE

CLOSED_SUBSPACE
    -> has(ORTHOGONAL_PROJECTION)

HILBERT_SPACE
    -> identified_with(DUAL_SPACE)
       via(RIESZ_REPRESENTATION)

OPERATOR
    -> has(ADJOINT)

SELF_ADJOINT_OPERATOR
    -> supports(SPECTRAL_ANALYSIS)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "Hilbert space" ~= "complete inner-product space",
    "complete inner product space" ~= "Hilbert space",
    "orthogonal projection" ~= "nearest-point projection onto a closed subspace",
    "orthonormal basis" ~= "complete orthonormal system spanning the Hilbert space densely",
    "Riesz representation" ~= "representation of continuous linear functionals by inner products",
    "L2 space" ~= "square-integrable function Hilbert space modulo almost-everywhere equality"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    hilbert_space_requires_inner_product_and_completeness: true,
    induced_norm_must_come_from_inner_product: true,
    every_hilbert_space_is_banach: true,
    not_every_banach_space_is_hilbert: true,
    projection_theorem_requires_closed_subspace: true,
    riesz_representation_depends_on_inner_product_convention: true,
    unbounded_operator_adjoints_require_domain_analysis: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    inner_product_norm_relation: 0.99,
    completeness_relation: 0.99,
    orthogonality_projection_relation: 0.99,
    riesz_representation_relation: 0.99,
    operator_spectral_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(vector_space)
    -> identify(inner_product)
    -> verify(inner_product_axioms)
    -> construct(induced_norm_and_metric)
    -> test(completeness)
    -> classify(as_hilbert_or_incomplete_inner_product_space)
    -> identify(orthogonal_subspaces_or_basis_if_relevant)
    -> construct(projection_if_requested)
    -> apply(riesz_representation_if_functional_present)
    -> analyse(adjoint_self_adjoint_or_compact_operator_if_relevant)
    -> verify(theorem_hypotheses)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed Hilbert-space semantic graph",
    primary_entity: HILBERT_SPACE,
    preserve: [
        vector_space,
        scalar_field,
        inner_product,
        induced_norm,
        metric,
        completeness,
        orthogonality,
        operator_domain
    ],
    infer_through: [
        orthonormal_bases,
        projections,
        riesz_representation,
        adjoint_operators,
        self_adjoint_operators,
        compact_operators,
        weak_convergence,
        spectral_theory
    ],
    avoid: [
        assuming_inner_product_space_is complete,
        assuming_arbitrary_norm_comes_from inner product,
        projecting_onto_nonclosed_subspace_as guaranteed nearest point,
        ignoring_complex_inner_product convention,
        ignoring_domains_of_unbounded operators,
        confusing_weak_and_norm convergence
    ]
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL(new_information):
    validate(new_information)
    -> identify(hilbert_space_context)
    -> compare(existing_relations)
    -> preserve(inner_product_completeness_and_operator_invariants)
    -> update(affected_entities)
    -> update(affected_edges)
    -> recompute(CONFIDENCE)
    -> regenerate(FLOW_HASH)

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + HILBERT_SPACE
    + INNER_PRODUCT
    + COMPLETENESS
    + ORTHONORMAL_BASIS
    + PROJECTION_THEOREM
    + RIESZ_REPRESENTATION_THEOREM
    + ADJOINT
    + SELF_ADJOINT_OPERATOR
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> HILBERT_SPACE
    -> VECTOR_SPACE_IDENTIFICATION
    -> INNER_PRODUCT_IDENTIFICATION
    -> INNER_PRODUCT_AXIOM_VALIDATION
    -> NORM_METRIC_CONSTRUCTION
    -> COMPLETENESS_ANALYSIS
    -> HILBERT_CLASSIFICATION
    -> ORTHOGONALITY_BASIS_ANALYSIS
    -> PROJECTION_ANALYSIS
    -> RIESZ_DUALITY_ANALYSIS
    -> OPERATOR_SPECTRAL_ANALYSIS
    -> HYPOTHESIS_VALIDATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW