Banach Space Ontology

flow "BanachSpaceOntology" v1.0

# ============================================================
# BANACH SPACE ONTOLOGY — FLOW ALGEBRA
# ============================================================

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

# ---------- CORE ENTITY ----------
BANACH_SPACE := {
    type: CompleteNormedVectorSpace,
    domain: FunctionalAnalysis,
    purpose: "Represent a normed vector space in which every Cauchy sequence converges to an element of the space."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    VECTOR_SPACE,
    NORM,
    NORMED_SPACE,
    METRIC,
    CAUCHY_SEQUENCE,
    COMPLETENESS,
    BANACH_SPACE,
    LINEAR_OPERATOR,
    BOUNDED_OPERATOR,
    DUAL_SPACE,
    CLOSED_SUBSPACE,
    QUOTIENT_SPACE,
    BANACH_ISOMORPHISM,
    HAHN_BANACH_THEOREM,
    OPEN_MAPPING_THEOREM,
    CLOSED_GRAPH_THEOREM
}

# ---------- BASIC STRUCTURE ----------
BANACH_SPACE
    := VECTOR_SPACE
       + NORM
       + COMPLETENESS

VECTOR_SPACE
    -> provides(ALGEBRAIC_STRUCTURE)

NORM
    -> provides(SIZE_AND_DISTANCE_STRUCTURE)

COMPLETENESS
    -> ensures(CAUCHY_LIMITS_REMAIN_IN_SPACE)

# ---------- NORM ----------
NORM
    := ||x||

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

NORM
    -> induces(METRIC)

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

# ---------- CAUCHY SEQUENCE ----------
sequence
    := (x_n)

CAUCHY_SEQUENCE
    iff

for_every epsilon > 0
exists N
such_that

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

CAUCHY_SEQUENCE
    -> internally_approaches(LIMIT_BEHAVIOUR)

# ---------- COMPLETENESS ----------
NORMED_SPACE X
    -> COMPLETE

iff

every_Cauchy_sequence_in(X)
    -> converges_to(element_of_X)

COMPLETENESS
    -> defining_property_of(BANACH_SPACE)

# ---------- CONVERGENCE ----------
x_n
    -> norm_converges_to(x)

iff

||x_n - x||
    -> 0

NORM_CONVERGENCE
    -> convergence_defined_by(NORM)

in Banach space:
    CAUCHY_SEQUENCE
        -> NORM_CONVERGENT_SEQUENCE

# ---------- INCOMPLETE NORMED SPACE ----------
NORMED_SPACE
    -/> necessarily(BANACH_SPACE)

if
    Cauchy_sequence_has_limit_outside_space
then

SPACE
    -> INCOMPLETE

INCOMPLETE_NORMED_SPACE
    -> may_have(COMPLETION)

# ---------- COMPLETION ----------
COMPLETION_OF_X
    := complete_normed_space(X_hat)

with
    isometric_embedding(
        X -> X_hat
    )

and
    image_of_X
        -> dense_in(X_hat)

COMPLETION
    -> smallest_complete_extension_up_to(ISOMETRIC_ISOMORPHISM)

# ---------- EXAMPLES ----------
FINITE_DIMENSIONAL_NORMED_SPACE
    -> BANACH_SPACE

R^n
    -> BANACH_SPACE
       under(any_norm)

C^n
    -> BANACH_SPACE
       under(any_norm)

C(K)
    -> BANACH_SPACE
       under(SUPREMUM_NORM)
       when(K_compact)

L^p
    -> BANACH_SPACE
       for(1 <= p <= infinity)

l^p
    -> BANACH_SPACE
       for(1 <= p <= infinity)

# ---------- HILBERT RELATION ----------
HILBERT_SPACE
    := COMPLETE_INNER_PRODUCT_SPACE

INNER_PRODUCT
    -> induces(NORM)

therefore

HILBERT_SPACE
    -> BANACH_SPACE

BANACH_SPACE
    -/> necessarily(HILBERT_SPACE)

# ---------- CLOSED SUBSPACE ----------
if
    X Banach
    AND M closed_linear_subspace_of(X)
then

M
    -> BANACH_SPACE
       under(RESTRICTED_NORM)

CLOSEDNESS
    -> preserves(COMPLETENESS_OF_SUBSPACE)

# ---------- QUOTIENT SPACE ----------
if
    X Banach
    AND M closed_linear_subspace
then

X/M
    -> BANACH_SPACE
       under(QUOTIENT_NORM)

QUOTIENT_NORM
    := inf(
        ||x+m|| :
        m in M
    )

CLOSED_SUBSPACE
    -> required_for(STANDARD_BANACH_QUOTIENT)

# ---------- PRODUCT SPACE ----------
if
    X and Y
        -> BANACH_SPACES
then

X x Y
    -> BANACH_SPACE
       under(STANDARD_PRODUCT_NORMS)

FINITE_PRODUCT
    -> preserves(COMPLETENESS)

# ---------- BOUNDED LINEAR OPERATOR ----------
T
    := linear_map(X,Y)

T
    -> bounded
       iff
       exists C >= 0
       such_that

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

for all x

between normed spaces:

BOUNDED_LINEAR
    <-> CONTINUOUS_LINEAR

# ---------- OPERATOR SPACE ----------
B(X,Y)
    := bounded_linear_operators_from(X,to=Y)

if
    Y Banach
then

B(X,Y)
    -> BANACH_SPACE
       under(OPERATOR_NORM)

OPERATOR_NORM
    := sup(
        ||T(x)|| :
        ||x|| <= 1
    )

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

X_star
    -> continuous_linear_functionals_on(X)

if
    X normed
then

X_star
    -> BANACH_SPACE
       under(DUAL_NORM)

DUAL_COMPLETENESS
    -> does_not_require(X_TO_BE_COMPLETE)

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

HAHN_BANACH
    -> supports(DUAL_SEPARATION)

# ---------- UNIFORM BOUNDEDNESS ----------
if
    X Banach
    AND family_of_operators
        -> pointwise_bounded
then

UNIFORM_BOUNDEDNESS_PRINCIPLE
    -> operator_norms_uniformly_bounded

BANACH_COMPLETENESS
    -> essential_hypothesis

# ---------- OPEN MAPPING ----------
if
    X,Y Banach
    AND T: X -> Y
        bounded_linear_surjective
then

T
    -> OPEN_MAP

OPEN_MAPPING_THEOREM
    -> fundamental_consequence_of(COMPLETENESS)

# ---------- BOUNDED INVERSE ----------
if
    X,Y Banach
    AND T: X -> Y
        bounded_linear_bijective
then

T_inverse
    -> bounded

BOUNDED_INVERSE_THEOREM
    -> guarantees(CONTINUOUS_INVERSE)

# ---------- CLOSED GRAPH ----------
GRAPH(T)
    := {
        (x,T(x)) :
        x in X
    }

if
    X,Y Banach
    AND T linear
    AND GRAPH(T) closed
then

T
    -> bounded

CLOSED_GRAPH_THEOREM
    -> derives(CONTINUITY_FROM_GRAPH_CLOSEDNESS)

# ---------- BANACH ISOMORPHISM ----------
BANACH_ISOMORPHISM
    := bounded_linear_bijection
       with_bounded_inverse

X
    ~= Y
       as(BANACH_SPACES)

if
    exists(BANACH_ISOMORPHISM)

BANACH_ISOMORPHISM
    -> preserves(TOPOLOGICAL_LINEAR_STRUCTURE)

# ---------- REFLEXIVITY ----------
CANONICAL_EMBEDDING
    := J: X -> X_double_star

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

X
    -> REFLEXIVE
       if
       J is surjective

REFLEXIVE_BANACH_SPACE
    -> has(STRONG_WEAK_COMPACTNESS_PROPERTIES)

# ---------- 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)

# ---------- FIXED POINT RELATION ----------
BANACH_FIXED_POINT_THEOREM
    -> applies_to(COMPLETE_METRIC_SPACE)

if
    T
        -> contraction
then

T
    -> has(UNIQUE_FIXED_POINT)

ITERATION
    -> converges_to(FIXED_POINT)

BANACH_SPACE
    -> complete_metric_space_under(norm_metric)

# ---------- DIFFERENTIAL EQUATIONS ----------
BANACH_SPACE
    -> provides(FUNCTION_SPACE_FRAMEWORK_FOR_ODEs_AND_PDEs)

FIXED_POINT_METHODS
    -> establish(EXISTENCE_AND_UNIQUENESS)

OPERATOR_EQUATIONS
    -> solved_in(BANACH_SPACES)

# ---------- APPLICATIONS ----------
BANACH_SPACE
    -> applied_to(FUNCTIONAL_ANALYSIS)
    -> applied_to(DIFFERENTIAL_EQUATIONS)
    -> applied_to(OPTIMISATION)
    -> applied_to(INTEGRAL_EQUATIONS)
    -> applied_to(NUMERICAL_ANALYSIS)
    -> applied_to(APPROXIMATION_THEORY)

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

NORMED_SPACE
    + COMPLETENESS
    -> BANACH_SPACE

BANACH_SPACE
    -> COMPLETE_METRIC_SPACE

HILBERT_SPACE
    -> BANACH_SPACE

CLOSED_SUBSPACE_OF_BANACH
    -> BANACH_SPACE

DUAL_SPACE_OF_NORMED_SPACE
    -> BANACH_SPACE

BANACH_COMPLETENESS
    -> enables(OPEN_MAPPING_THEOREM)
    -> enables(CLOSED_GRAPH_THEOREM)
    -> enables(UNIFORM_BOUNDEDNESS_PRINCIPLE)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "Banach space" ~= "complete normed vector space",
    "complete normed space" ~= "Banach space",
    "Banach-space completeness" ~= "every norm-Cauchy sequence converges within the space",
    "Banach isomorphism" ~= "bounded linear bijection with bounded inverse",
    "completion" ~= "complete normed extension containing the original space densely",
    "operator space B(X,Y)" ~= "space of bounded linear operators from X to Y"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    banach_space_requires_norm_and_completeness: true,
    normed_space_need_not_be_complete: true,
    hilbert_space_is_banach_but_banach_need_not_be_hilbert: true,
    closed_subspace_of_banach_is_banach: true,
    quotient_by_closed_subspace_is_banach: true,
    dual_of_any_normed_space_is_banach: true,
    major_banach_theorems_require_their_specific_hypotheses: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    completeness_relation: 0.99,
    completion_relation: 0.99,
    operator_space_relation: 0.99,
    dual_space_relation: 0.99,
    fundamental_theorem_relations: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(vector_space)
    -> identify(norm)
    -> verify(norm_axioms)
    -> construct(norm_metric)
    -> test(cauchy_sequence_behaviour)
    -> determine(completeness)
    -> classify(as_banach_or_incomplete)
    -> identify(subspace_dual_quotient_or_operator_structure_if_relevant)
    -> select(applicable_banach_space_theorem)
    -> verify(theorem_hypotheses)
    -> analyse(convergence_or_operator_consequences)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed Banach-space semantic graph",
    primary_entity: BANACH_SPACE,
    preserve: [
        vector_space,
        scalar_field,
        norm,
        metric,
        cauchy_sequences,
        completeness,
        subspace_structure,
        operator_domains
    ],
    infer_through: [
        completions,
        bounded_operators,
        dual_spaces,
        quotient_spaces,
        weak_convergence,
        open_mapping,
        closed_graph,
        uniform_boundedness
    ],
    avoid: [
        assuming_normed_space_is complete,
        confusing_banach_with_hilbert,
        assuming_subspace_is complete_without closedness,
        forming_standard_banach_quotient_by nonclosed subspace,
        ignoring_operator_space codomain completeness,
        applying_banach_theorems_without hypotheses
    ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + BANACH_SPACE
    + NORM
    + CAUCHY_SEQUENCE
    + COMPLETENESS
    + COMPLETION_OF_X
    + BOUNDED_LINEAR_OPERATOR
    + DUAL_SPACE
    + OPEN_MAPPING_THEOREM
    + CLOSED_GRAPH_THEOREM
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> BANACH_SPACE
    -> VECTOR_SPACE_IDENTIFICATION
    -> NORM_IDENTIFICATION
    -> NORM_AXIOM_VALIDATION
    -> METRIC_CONSTRUCTION
    -> CAUCHY_ANALYSIS
    -> COMPLETENESS_ANALYSIS
    -> BANACH_CLASSIFICATION
    -> SUBSPACE_DUAL_OPERATOR_ANALYSIS
    -> THEOREM_SELECTION
    -> HYPOTHESIS_VALIDATION
    -> CONVERGENCE_OPERATOR_INTERPRETATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW