Functional Analysis Ontology

flow "FunctionalAnalysisOntology" v1.0

# ============================================================
# FUNCTIONAL ANALYSIS ONTOLOGY — FLOW ALGEBRA
# ============================================================

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

# ---------- CORE ENTITY ----------
FUNCTIONAL_ANALYSIS := {
    type: MathematicalDiscipline,
    domain: Mathematics,
    parent: MathematicalAnalysis,
    purpose: "Formal study of function spaces, normed and inner-product spaces, continuous linear operators, duality, convergence and spectral structure, especially in infinite-dimensional settings."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    VECTOR_SPACE,
    NORMED_SPACE,
    BANACH_SPACE,
    INNER_PRODUCT_SPACE,
    HILBERT_SPACE,
    TOPOLOGICAL_VECTOR_SPACE,
    LINEAR_OPERATOR,
    BOUNDED_LINEAR_OPERATOR,
    LINEAR_FUNCTIONAL,
    DUAL_SPACE,
    ADJOINT_OPERATOR,
    COMPACT_OPERATOR,
    WEAK_CONVERGENCE,
    SPECTRUM,
    RESOLVENT
}

# ---------- ONTOLOGICAL STRUCTURE ----------
FUNCTIONAL_ANALYSIS
    -> branch_of(MATHEMATICAL_ANALYSIS)
    -> generalises(FINITE_DIMENSIONAL_LINEAR_ALGEBRA)
    -> studies(FUNCTION_SPACES)
    -> studies(OPERATORS)
    -> uses(TOPOLOGY)
    -> uses(MEASURE_THEORY)
    -> supports(DIFFERENTIAL_EQUATION_ANALYSIS)

# ---------- VECTOR SPACE ----------
VECTOR_SPACE
    := set_with(
        VECTOR_ADDITION,
        SCALAR_MULTIPLICATION
    )

VECTOR_SPACE
    -> defined_over(SCALAR_FIELD)

SCALAR_FIELD
    -> commonly(REAL_NUMBERS_OR_COMPLEX_NUMBERS)

FUNCTIONS
    -> may_be(VECTORS_IN_FUNCTION_SPACE)

# ---------- NORM ----------
NORM
    := mapping(
        VECTOR_SPACE -> NONNEGATIVE_REAL_NUMBERS
    )

NORM(x)
    := ||x||

NORM_PROPERTIES := {
    ||x|| >= 0,
    ||x|| = 0 iff x = 0,
    ||alpha*x|| = abs(alpha)*||x||,
    ||x+y|| <= ||x|| + ||y||
}

# ---------- NORMED SPACE ----------
NORMED_SPACE
    := VECTOR_SPACE + NORM

NORM
    -> induces(METRIC)

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

METRIC
    -> induces(TOPOLOGY)

# ---------- CAUCHY SEQUENCE ----------
CAUCHY_SEQUENCE
    -> terms_become_arbitrarily_close

for every epsilon > 0
exists N
such_that

m,n >= N
    -> ||x_m - x_n|| < epsilon

CAUCHY
    -> central_to(COMPLETENESS)

# ---------- BANACH SPACE ----------
BANACH_SPACE
    := NORMED_SPACE
       + COMPLETENESS

COMPLETENESS
    -> every_Cauchy_sequence_converges_to_element_of_space

BANACH_SPACE
    -> stable_under(MANY_LIMIT_PROCESSES)

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

INNER_PRODUCT
    -> satisfies(
        LINEARITY_OR_SESQUILINEARITY,
        CONJUGATE_SYMMETRY,
        POSITIVE_DEFINITENESS
    )

NORM_FROM_INNER_PRODUCT
    := sqrt(<x,x>)

# ---------- HILBERT SPACE ----------
HILBERT_SPACE
    := INNER_PRODUCT_SPACE
       + COMPLETENESS

HILBERT_SPACE
    -> BANACH_SPACE_WITH_INNER_PRODUCT_NORM

HILBERT_SPACE
    -> supports(ORTHOGONALITY)
    -> supports(PROJECTION)
    -> supports(ORTHONORMAL_EXPANSIONS)

# ---------- TOPOLOGICAL VECTOR SPACE ----------
TOPOLOGICAL_VECTOR_SPACE
    := VECTOR_SPACE
       + TOPOLOGY

VECTOR_ADDITION
    -> continuous

SCALAR_MULTIPLICATION
    -> continuous

TOPOLOGICAL_VECTOR_SPACE
    -> generalises(NORMED_SPACE)

# ---------- LINEAR OPERATOR ----------
LINEAR_OPERATOR
    := T: X -> Y

T(alpha*x + beta*y)
    =
alpha*T(x)
+
beta*T(y)

DOMAIN(T)
    -> X

CODOMAIN(T)
    -> Y

# ---------- BOUNDED LINEAR OPERATOR ----------
T
    -> bounded
       iff
       exists C >= 0
       such_that

||T(x)||
    <= C||x||

for_all x

for linear operators between normed spaces:

BOUNDED
    <-> CONTINUOUS

# ---------- OPERATOR NORM ----------
||T||
    := sup(
        ||T(x)|| :
        ||x|| <= 1
    )

OPERATOR_NORM
    -> finite_for(BOUNDED_OPERATOR)

BOUNDED_OPERATORS(X,Y)
    -> normed_space

if
    Y Banach
then

BOUNDED_OPERATORS(X,Y)
    -> Banach_space

# ---------- LINEAR FUNCTIONAL ----------
LINEAR_FUNCTIONAL
    := f: X -> FIELD

f(alpha*x + beta*y)
    =
alpha*f(x)
+
beta*f(y)

CONTINUOUS_LINEAR_FUNCTIONAL
    -> BOUNDED_LINEAR_FUNCTIONAL

# ---------- DUAL SPACE ----------
DUAL_SPACE
    := X_star

X_star
    -> set_of(ALL_CONTINUOUS_LINEAR_FUNCTIONALS_ON_X)

DUAL_SPACE
    -> normed_space_under(DUAL_NORM)

DUAL_NORM(f)
    := sup(
        abs(f(x)) :
        ||x|| <= 1
    )

# ---------- BIDUAL ----------
BIDUAL
    := X_double_star

CANONICAL_MAP
    := J: X -> X_double_star

J(x)(f)
    := f(x)

REFLEXIVE_SPACE
    -> J_is_surjective

REFLEXIVITY
    -> important_to(WEAK_COMPACTNESS)

# ---------- ADJOINT OPERATOR ----------
for bounded operator
    T: X -> Y

BANACH_ADJOINT
    := T_star: Y_star -> X_star

defined_by

T_star(g)
    := g composed_with T

in Hilbert spaces:
    ADJOINT_OPERATOR
        -> represented_through(INNER_PRODUCT_RELATION)

# ---------- HILBERT ADJOINT ----------
for densely defined operator T:

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

for appropriate x,y

ADJOINT_DOMAIN
    -> essential_for(UNBOUNDED_OPERATORS)

# ---------- SELF-ADJOINT OPERATOR ----------
T
    -> SELF_ADJOINT
       if
       T = T_star
       including(DOMAIN_EQUALITY_FOR_UNBOUNDED_CASE)

SELF_ADJOINT_OPERATOR
    -> has(REAL_SPECTRUM)

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

equivalently in metric-space setting:
    every_bounded_sequence
        -> has_subsequence_whose_images_converge

COMPACT_OPERATOR
    -> spectral_structure_resembles(FINITE_DIMENSIONAL_OPERATOR)

# ---------- FINITE RANK ----------
FINITE_RANK_OPERATOR
    -> range_has(FINITE_DIMENSION)

FINITE_RANK_OPERATOR
    -> COMPACT

COMPACT_OPERATOR
    -> may_be_approximated_by(FINITE_RANK_OPERATORS)
       under(SPACE_SPECIFIC_CONDITIONS)

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

iff

f(x_n)
    -> f(x)

for_all
    f in X_star

NORM_CONVERGENCE
    -> implies(WEAK_CONVERGENCE)

WEAK_CONVERGENCE
    -/> generally_implies(NORM_CONVERGENCE)

# ---------- WEAK STAR CONVERGENCE ----------
f_n
    -> weak_star_converges_to(f)

iff

f_n(x)
    -> f(x)

for_all
    x in X

WEAK_STAR_TOPOLOGY
    -> coarsest_topology_making(EVALUATION_MAPS_CONTINUOUS)

# ---------- BANACH-ALAOGLU ----------
BANACH_ALAOGLU_THEOREM
    -> closed_unit_ball_of_dual
    -> compact_in(WEAK_STAR_TOPOLOGY)

BANACH_ALAOGLU
    -> foundational_compactness_result

# ---------- HAHN-BANACH ----------
HAHN_BANACH_THEOREM
    -> extends(BOUNDED_LINEAR_FUNCTIONALS)
    -> preserves(NORM_OR_DOMINATION_BOUND)

HAHN_BANACH
    -> separates(POINTS_AND_SUBSPACES)
       in(APPROPRIATE_FORMS)

# ---------- UNIFORM BOUNDEDNESS ----------
UNIFORM_BOUNDEDNESS_PRINCIPLE
    -> pointwise_bounded_family_of_bounded_operators
    -> uniformly_bounded_in_operator_norm
       when(DOMAIN_IS_BANACH)

# ---------- OPEN MAPPING ----------
OPEN_MAPPING_THEOREM
    -> surjective_bounded_linear_map_between_Banach_spaces
    -> OPEN_MAP

# ---------- BOUNDED INVERSE ----------
if
    T
        := bounded_bijective_linear_operator
    between Banach spaces
then

T_inverse
    -> bounded

BOUNDED_INVERSE_THEOREM
    -> follows_from(OPEN_MAPPING_THEOREM)

# ---------- CLOSED GRAPH ----------
if
    T: X -> Y
    linear
    between Banach spaces
    and graph(T) closed
then

T
    -> bounded

CLOSED_GRAPH_THEOREM
    -> operator_continuity_from(GRAPH_CLOSEDNESS)

# ---------- SPECTRUM ----------
for bounded operator T on Banach space:

SPECTRUM(T)
    := {
        lambda :
        T - lambda*I
        is_not_boundedly_invertible
    }

SPECTRUM
    -> may_include(EIGENVALUES)
    -> may_include(CONTINUOUS_SPECTRAL_VALUES)
    -> may_include(RESIDUAL_SPECTRAL_VALUES)

# ---------- RESOLVENT ----------
RESOLVENT_SET(T)
    := FIELD minus SPECTRUM(T)

for
    lambda in RESOLVENT_SET:

R(lambda,T)
    := (
        T - lambda*I
    )^(-1)

RESOLVENT_OPERATOR
    -> analytic_in(lambda)
       on(RESOLVENT_SET)

# ---------- SPECTRAL RADIUS ----------
SPECTRAL_RADIUS(T)
    := sup(
        abs(lambda) :
        lambda in SPECTRUM(T)
    )

SPECTRAL_RADIUS_FORMULA:
    r(T)
    =
    limit_n_to_infinity(
        ||T^n||^(1/n)
    )

for bounded operator on complex Banach space

# ---------- RIESZ REPRESENTATION ----------
for Hilbert space H:

every
    continuous_linear_functional f
    -> represented_by(unique y in H)

such_that

f(x)
    = <x,y>

subject_to
    INNER_PRODUCT_CONVENTION

RIESZ_REPRESENTATION
    -> identifies(H_WITH_ITS_DUAL_UP_TO_CONVENTION)

# ---------- ORTHOGONAL PROJECTION ----------
for
    closed_subspace M
    of Hilbert space H

each x in H
    -> unique decomposition

x
    = P_M x
      + z

where

P_M x in M
z in M_perp

PROJECTION_THEOREM
    -> gives(BEST_APPROXIMATION_IN_CLOSED_SUBSPACE)

# ---------- FUNCTION SPACES ----------
FUNCTION_SPACES := {
    C(K),
    Lp,
    l_p,
    SOBOLEV_SPACES,
    DISTRIBUTION_SPACES
}

FUNCTIONAL_ANALYSIS
    -> provides(STRUCTURE_FOR_FUNCTION_SPACES)

# ---------- DIFFERENTIAL EQUATIONS ----------
DIFFERENTIAL_OPERATOR
    -> operator_between(FUNCTION_SPACES)

PDE
    -> operator_equation

L[u]
    = f

FUNCTIONAL_ANALYSIS
    -> supports(
        EXISTENCE,
        UNIQUENESS,
        STABILITY,
        SPECTRAL_ANALYSIS
    )

# ---------- APPLICATIONS ----------
FUNCTIONAL_ANALYSIS
    -> applied_to(PDEs)
    -> applied_to(QUANTUM_MECHANICS)
    -> applied_to(FOURIER_ANALYSIS)
    -> applied_to(OPTIMISATION)
    -> applied_to(CONTROL_THEORY)
    -> applied_to(PROBABILITY)
    -> applied_to(NUMERICAL_ANALYSIS)

# ---------- RELATION GRAPH ----------
VECTOR_SPACE
    + NORM
    -> NORMED_SPACE

NORMED_SPACE
    + COMPLETENESS
    -> BANACH_SPACE

INNER_PRODUCT_SPACE
    + COMPLETENESS
    -> HILBERT_SPACE

NORMED_SPACE
    -> generates(DUAL_SPACE)

LINEAR_OPERATOR
    + BOUNDEDNESS
    -> CONTINUOUS_LINEAR_OPERATOR

OPERATOR
    -> has(SPECTRUM)

HILBERT_SPACE
    -> supports(ADJOINT_AND_PROJECTION)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "functional analysis" ~= "analysis of infinite-dimensional vector and function spaces and operators",
    "Banach space" ~= "complete normed vector space",
    "Hilbert space" ~= "complete inner-product space",
    "dual space" ~= "space of continuous linear functionals",
    "bounded linear operator" ~= "continuous linear operator between normed spaces",
    "weak convergence" ~= "convergence against every continuous linear functional"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    completeness_is_relative_to_metric_or_norm: true,
    bounded_linear_operator_is_continuous_on_normed_spaces: true,
    weak_convergence_does_not_generally_imply_norm_convergence: true,
    spectrum_is_not_only_point_spectrum: true,
    unbounded_operator_adjoints_require_domain_analysis: true,
    self_adjointness_requires_domain_equality_in_unbounded_case: true,
    theorem_applications_require_exact_hypotheses: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    ontology_structure: 0.99,
    banach_hilbert_relations: 0.99,
    operator_relation: 0.99,
    duality_relation: 0.99,
    convergence_relation: 0.99,
    spectral_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(space_operator_or_functional)
    -> identify(algebraic_structure)
    -> identify(norm_inner_product_or_topology)
    -> test(completeness_if_relevant)
    -> identify(operator_domain_and_codomain)
    -> test(linearity_boundedness_continuity)
    -> construct(dual_adjoint_or_bidual_if_required)
    -> classify(convergence_mode)
    -> analyse(compactness_or_spectrum_if_requested)
    -> select(applicable_theorem)
    -> verify(all_hypotheses)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed functional-analysis ontology graph",
    primary_entity: FUNCTIONAL_ANALYSIS,
    preserve: [
        vector_space,
        scalar_field,
        norm,
        inner_product,
        topology,
        completeness,
        operator_domain,
        operator_codomain
    ],
    infer_through: [
        normed_spaces,
        banach_spaces,
        hilbert_spaces,
        dual_spaces,
        adjoint_operators,
        compact_operators,
        weak_topologies,
        spectra
    ],
    avoid: [
        assuming_normed_space_is Banach,
        assuming_Banach_space_is Hilbert,
        treating_all_spectral_values_as eigenvalues,
        ignoring_domains_of_unbounded operators,
        confusing_weak_and_norm convergence,
        applying_theorems_without hypotheses
    ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + FUNCTIONAL_ANALYSIS
    + NORMED_SPACE
    + BANACH_SPACE
    + HILBERT_SPACE
    + BOUNDED_LINEAR_OPERATOR
    + DUAL_SPACE
    + ADJOINT_OPERATOR
    + WEAK_CONVERGENCE
    + SPECTRUM
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> FUNCTIONAL_ANALYSIS
    -> SPACE_OPERATOR_FUNCTIONAL_IDENTIFICATION
    -> ALGEBRAIC_STRUCTURE_ANALYSIS
    -> NORM_INNER_PRODUCT_TOPOLOGY_ANALYSIS
    -> COMPLETENESS_ANALYSIS
    -> OPERATOR_DOMAIN_ANALYSIS
    -> LINEARITY_BOUNDEDNESS_ANALYSIS
    -> DUAL_ADJOINT_ANALYSIS
    -> CONVERGENCE_ANALYSIS
    -> COMPACTNESS_SPECTRAL_ANALYSIS
    -> THEOREM_SELECTION
    -> HYPOTHESIS_VALIDATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW