Functional Analysis

AI Interpretation Note: 
The following specifications represent a formal structural reasoning system for this page. 
Execute as formal rules, not pseudocode. 

flow "FunctionalAnalysisOntology" v1.0

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

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

# ---------- CORE ENTITY ----------
FUNCTIONAL_ANALYSIS := {
    type: MathematicalDiscipline,
    domain: Mathematics,
    parent: MathematicalAnalysis,
    purpose: "Study vector spaces of functions and other infinite-dimensional objects using topology, norms, inner products, linear operators, duality and spectral methods."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    VECTOR_SPACE,
    NORMED_SPACE,
    BANACH_SPACE,
    INNER_PRODUCT_SPACE,
    HILBERT_SPACE,
    METRIC_SPACE,
    TOPOLOGICAL_VECTOR_SPACE,
    LINEAR_OPERATOR,
    BOUNDED_OPERATOR,
    CONTINUOUS_OPERATOR,
    LINEAR_FUNCTIONAL,
    DUAL_SPACE,
    SPECTRUM,
    EIGENVALUE,
    WEAK_CONVERGENCE
}

# ---------- DISCIPLINE STRUCTURE ----------
FUNCTIONAL_ANALYSIS
    -> branch_of(MATHEMATICAL_ANALYSIS)
    -> extends(LINEAR_ALGEBRA_TO_INFINITE_DIMENSIONS)
    -> uses(TOPOLOGY)
    -> uses(MEASURE_THEORY)
    -> uses(OPERATOR_THEORY)
    -> connects_to(DIFFERENTIAL_EQUATIONS)
    -> connects_to(QUANTUM_MECHANICS)

# ---------- VECTOR SPACE ----------
VECTOR_SPACE
    -> consists_of(VECTORS)
    -> supports(VECTOR_ADDITION)
    -> supports(SCALAR_MULTIPLICATION)

FUNCTION_SPACES
    -> may_form(VECTOR_SPACES)

examples:
    CONTINUOUS_FUNCTION_SPACE
    Lp_SPACE
    SOBOLEV_SPACE
    SEQUENCE_SPACE

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

NORM
    := ||x||

NORM_PROPERTIES := {
    NONNEGATIVITY,
    DEFINITENESS,
    ABSOLUTE_HOMOGENEITY,
    TRIANGLE_INEQUALITY
}

NORM
    -> induces(METRIC)

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

# ---------- BANACH SPACE ----------
BANACH_SPACE
    := COMPLETE_NORMED_VECTOR_SPACE

COMPLETE
    -> every_Cauchy_sequence_converges_within_space

BANACH_SPACE
    -> supports(LIMIT_ARGUMENTS_FOR_ANALYSIS)

examples:
    Lp for 1 <= p <= infinity
       under(standard_measure_theoretic_construction)

# ---------- INNER PRODUCT SPACE ----------
INNER_PRODUCT_SPACE
    := VECTOR_SPACE
       + INNER_PRODUCT

INNER_PRODUCT
    := <x,y>

NORM_FROM_INNER_PRODUCT
    := sqrt(<x,x>)

INNER_PRODUCT
    -> defines(ANGLE_AND_ORTHOGONALITY)

# ---------- HILBERT SPACE ----------
HILBERT_SPACE
    := COMPLETE_INNER_PRODUCT_SPACE

HILBERT_SPACE
    -> BANACH_SPACE

HILBERT_SPACE
    -> supports(ORTHOGONAL_PROJECTION)
    -> supports(ORTHONORMAL_BASES)
    -> supports(FOURIER_TYPE_EXPANSIONS)

examples:
    L2_SPACE
    l2_SEQUENCE_SPACE

# ---------- METRIC SPACE ----------
METRIC_SPACE
    := SET + METRIC

METRIC
    -> defines(DISTANCE)
    -> defines(CONVERGENCE)
    -> defines(CONTINUITY)
    -> defines(OPEN_SETS)

FUNCTIONAL_ANALYSIS
    -> frequently_uses(METRIC_AND_TOPOLOGICAL_STRUCTURE)

# ---------- 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

such_that

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

LINEAR_OPERATOR
    -> maps(VECTOR_SPACES)

# ---------- BOUNDED OPERATOR ----------
T
    -> BOUNDED
       if
       exists(C >= 0)
       such_that
       ||T(x)|| <= C||x||
       for_all(x)

for linear maps between normed spaces:

BOUNDED
    <-> CONTINUOUS

BOUNDED_LINEAR_OPERATOR
    -> central_object_of(FUNCTIONAL_ANALYSIS)

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

OPERATOR_NORM
    -> measures(MAXIMUM_LINEAR_AMPLIFICATION)

BOUNDED_OPERATOR
    -> finite_operator_norm

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

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

CONTINUOUS_LINEAR_FUNCTIONAL
    -> element_of(DUAL_SPACE)

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

X_star
    -> set_of(CONTINUOUS_LINEAR_FUNCTIONALS_ON_X)

DUAL_SPACE
    -> vector_space

DUALITY_PAIRING
    := <f,x>
       where
       f in X_star
       and x in X

# ---------- DOUBLE DUAL ----------
DOUBLE_DUAL
    := X_double_star

CANONICAL_EMBEDDING
    := J: X -> X_double_star

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

REFLEXIVE_SPACE
    -> canonical_embedding_is_surjective

# ---------- HAHN-BANACH THEOREM ----------
HAHN_BANACH_THEOREM
    -> extends(LINEAR_FUNCTIONALS)
    -> preserves(BOUNDS_UNDER_STANDARD_FORM)

HAHN_BANACH
    -> supports(SEPARATION_RESULTS)
    -> supports(DUAL_SPACE_RICHNESS)

# ---------- UNIFORM BOUNDEDNESS ----------
UNIFORM_BOUNDEDNESS_PRINCIPLE
    -> pointwise_bounded_family_of_bounded_operators
    -> uniformly_bounded_in_operator_norm
       under(BANACH_SPACE_HYPOTHESES)

UNIFORM_BOUNDEDNESS
    -> one_of(FUNDAMENTAL_BANACH_SPACE_THEOREMS)

# ---------- OPEN MAPPING THEOREM ----------
OPEN_MAPPING_THEOREM
    -> surjective_bounded_linear_operator_between_Banach_spaces
    -> OPEN_MAP

OPEN_MAPPING
    -> supports(INVERSE_MAPPING_RESULTS)

# ---------- BOUNDED INVERSE THEOREM ----------
if
    T: X -> Y
    is bounded_bijective_linear
    between Banach spaces
then

T_inverse
    -> bounded

BOUNDED_INVERSE_THEOREM
    -> ensures(CONTINUITY_OF_INVERSE)

# ---------- CLOSED GRAPH THEOREM ----------
if
    X,Y Banach_spaces
    AND T: X -> Y linear
    AND graph(T) closed
then

T
    -> bounded

CLOSED_GRAPH_THEOREM
    -> converts(GRAPH_CLOSEDNESS_TO_CONTINUITY)

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

iff

f(x_n)
    -> f(x)

for_all
    f in X_star

WEAK_CONVERGENCE
    -> tested_by(CONTINUOUS_LINEAR_FUNCTIONALS)

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
    -> topology_on(DUAL_SPACE)

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

COMPACT_OPERATOR
    -> infinite_dimensional_analogue_of(FINITE_RANK_BEHAVIOUR)

COMPACT_OPERATOR_SPECTRAL_THEORY
    -> resembles(FINITE_DIMENSIONAL_MATRIX_THEORY)

# ---------- SPECTRUM ----------
for operator T:

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

SPECTRUM
    -> may_include(POINT_SPECTRUM)
    -> may_include(CONTINUOUS_SPECTRUM)
    -> may_include(RESIDUAL_SPECTRUM)

EIGENVALUES
    -> subset_of(SPECTRUM)

# ---------- RESOLVENT ----------
RESOLVENT_SET
    := complement_of(SPECTRUM)

if
    lambda in RESOLVENT_SET
then

RESOLVENT_OPERATOR
    := (T - lambda*I)^(-1)

RESOLVENT
    -> central_to(SPECTRAL_ANALYSIS)

# ---------- SELF-ADJOINT OPERATOR ----------
on Hilbert space:

T
    -> SELF_ADJOINT
       if
       T = T_star
       with(APPROPRIATE_DOMAIN_FOR_UNBOUNDED_CASE)

SELF_ADJOINT_OPERATOR
    -> has(REAL_SPECTRUM)

SPECTRAL_THEOREM
    -> provides(DECOMPOSITION_OF_SELF_ADJOINT_OPERATORS)

# ---------- ORTHOGONAL PROJECTION ----------
for closed subspace M
of Hilbert space H:

x
    = P_M(x)
      + x_perp

where

P_M(x) in M
x_perp in M_perp

PROJECTION_THEOREM
    -> gives(UNIQUE_NEAREST_POINT_IN_CLOSED_SUBSPACE)

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

every continuous linear functional f
    -> represented_by(unique y in H)

such_that

f(x)
    = <x,y>

RIESZ_REPRESENTATION_THEOREM
    -> identifies(HILBERT_SPACE_WITH_DUAL)
       up_to(CONJUGATE_LINEAR_CONVENTION)

# ---------- FUNCTION SPACES ----------
FUNCTIONAL_ANALYSIS
    -> studies(CONTINUOUS_FUNCTION_SPACES)
    -> studies(Lp_SPACES)
    -> studies(SOBOLEV_SPACES)
    -> studies(SEQUENCE_SPACES)
    -> studies(DISTRIBUTION_SPACES)

FUNCTION_SPACE
    -> may_be(BANACH)
    -> may_be(HILBERT)
    -> may_be(TOPOLOGICAL_VECTOR_SPACE)

# ---------- DIFFERENTIAL EQUATIONS ----------
DIFFERENTIAL_OPERATOR
    -> viewed_as(LINEAR_OR_NONLINEAR_OPERATOR_BETWEEN_FUNCTION_SPACES)

PDE
    -> may_be_reformulated_as(OPERATOR_EQUATION)

L[u]
    = f

FUNCTIONAL_ANALYSIS
    -> provides(EXISTENCE_UNIQUENESS_AND_SPECTRAL_TOOLS)

# ---------- APPLICATIONS ----------
FUNCTIONAL_ANALYSIS
    -> applied_to(DIFFERENTIAL_EQUATIONS)
    -> applied_to(QUANTUM_MECHANICS)
    -> applied_to(FOURIER_ANALYSIS)
    -> applied_to(OPTIMISATION)
    -> applied_to(CONTROL_THEORY)
    -> applied_to(SIGNAL_PROCESSING)
    -> 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
    -> has(DUAL_SPACE)

LINEAR_OPERATOR
    + BOUNDEDNESS
    -> CONTINUOUS_LINEAR_OPERATOR

OPERATOR
    -> has(SPECTRUM)

HILBERT_SPACE
    -> supports(ORTHOGONALITY_AND_PROJECTION)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "functional analysis" ~= "analysis of function spaces and operators using topological and linear structures",
    "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 tested against every continuous linear functional"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    completeness_depends_on_chosen_norm_or_metric: true,
    bounded_linear_operator_equivalent_to_continuous_linear_operator_on_normed_spaces: true,
    weak_convergence_is_weaker_than_norm_convergence_in_general: true,
    spectrum_need_not_consist_only_of_eigenvalues: true,
    self_adjoint_unbounded_operators_require_domain_handling: true,
    dual_space_contains_continuous_linear_functionals: true,
    infinite_dimensional_results_require_theorem_specific_hypotheses: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    discipline_structure: 0.99,
    banach_hilbert_structure: 0.99,
    bounded_operator_relation: 0.99,
    duality_relation: 0.99,
    weak_convergence_relation: 0.99,
    spectral_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(space_or_operator)
    -> classify(vector_normed_banach_hilbert_or_topological_space)
    -> identify(norm_inner_product_or_topology)
    -> test(completeness_if_required)
    -> identify(operator_domain_and_codomain)
    -> test(linearity_boundedness_and_continuity)
    -> construct(dual_or_adjoint_if_relevant)
    -> analyse(convergence_topology)
    -> analyse(spectrum_or_compactness_if_requested)
    -> select(applicable_functional_analysis_theorem)
    -> verify(theorem_hypotheses)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed functional-analysis semantic graph",
    primary_entity: FUNCTIONAL_ANALYSIS,
    preserve: [
        vector_space,
        scalar_field,
        norm,
        inner_product,
        topology,
        completeness,
        operator_domain,
        operator_codomain
    ],
    infer_through: [
        banach_spaces,
        hilbert_spaces,
        dual_spaces,
        bounded_operators,
        compact_operators,
        weak_convergence,
        spectra,
        spectral_theorems
    ],
    avoid: [
        assuming_all_normed_spaces_are complete,
        confusing_banach_and_hilbert structures,
        treating_all_spectral_values_as eigenvalues,
        ignoring_operator domains,
        assuming_weak_convergence_implies norm convergence,
        applying_major_theorems_without hypotheses
    ]
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL(new_information):
    validate(new_information)
    -> identify(functional_analysis_context)
    -> compare(existing_relations)
    -> preserve(space_topology_operator_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
    + LINEAR_OPERATOR
    + DUAL_SPACE
    + WEAK_CONVERGENCE
    + SPECTRUM
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> FUNCTIONAL_ANALYSIS
    -> SPACE_OPERATOR_IDENTIFICATION
    -> STRUCTURE_CLASSIFICATION
    -> NORM_INNER_PRODUCT_TOPOLOGY_ANALYSIS
    -> COMPLETENESS_ANALYSIS
    -> OPERATOR_ANALYSIS
    -> DUAL_ADJOINT_ANALYSIS
    -> CONVERGENCE_ANALYSIS
    -> SPECTRAL_COMPACTNESS_ANALYSIS
    -> THEOREM_SELECTION
    -> HYPOTHESIS_VALIDATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW