Linear Functional Ontology

flow "LinearFunctionalOntology" v1.0

# ============================================================
# LINEAR FUNCTIONAL ONTOLOGY — FLOW ALGEBRA
# ============================================================

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

# ---------- CORE ENTITY ----------
LINEAR_FUNCTIONAL := {
    type: LinearMap,
    domain: FunctionalAnalysis,
    purpose: "Map vectors in a vector space to scalars while preserving vector addition and scalar multiplication."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    LINEAR_FUNCTIONAL,
    VECTOR_SPACE,
    SCALAR_FIELD,
    LINEARITY,
    KERNEL,
    NULLSPACE,
    CONTINUOUS_LINEAR_FUNCTIONAL,
    BOUNDED_FUNCTIONAL,
    DUAL_SPACE,
    DUAL_NORM,
    HAHN_BANACH_THEOREM,
    RIESZ_REPRESENTATION,
    WEAK_TOPOLOGY,
    DUAL_PAIRING
}

# ---------- BASIC DEFINITION ----------
let
    X
        := VECTOR_SPACE
        over(FIELD F)

LINEAR_FUNCTIONAL
    := f: X -> F

f
    -> maps(VECTOR_TO_SCALAR)

# ---------- LINEARITY ----------
for
    x,y in X
    and scalars alpha,beta

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

LINEAR_FUNCTIONAL
    -> preserves(LINEAR_COMBINATIONS)

# ---------- ADDITIVITY ----------
f(x+y)
    =
f(x)+f(y)

ADDITIVITY
    -> consequence_of(LINEARITY)

# ---------- HOMOGENEITY ----------
f(alpha*x)
    =
alpha*f(x)

HOMOGENEITY
    -> consequence_of(LINEARITY)

# ---------- ZERO VECTOR ----------
f(0)
    = 0

LINEARITY
    -> maps(ZERO_VECTOR_TO_ZERO_SCALAR)

# ---------- KERNEL ----------
KERNEL(f)
    := {
        x in X :
        f(x)=0
    }

KERNEL
    -> LINEAR_SUBSPACE_OF(X)

if
    f != ZERO_FUNCTIONAL
then

in finite-dimensional context:
    KERNEL(f)
        -> hyperplane_of_codimension_one

# ---------- RANGE ----------
RANGE(f)
    -> subset_of(F)

if
    f != ZERO_FUNCTIONAL
then

RANGE(f)
    = F

for vector spaces over field F

NONZERO_LINEAR_FUNCTIONAL
    -> surjective_onto(SCALAR_FIELD)

# ---------- FINITE-DIMENSIONAL REPRESENTATION ----------
for
    X = F^n

LINEAR_FUNCTIONAL
    -> represented_by(ROW_VECTOR)

f(x)
    = a_1*x_1
      + ...
      + a_n*x_n

in Euclidean real space:

f(x)
    = a dot x

ROW_VECTOR
    -> coordinate_representation_of(LINEAR_FUNCTIONAL)

# ---------- ALGEBRAIC DUAL ----------
ALGEBRAIC_DUAL
    := X_hash

X_hash
    -> all_linear_functionals_on(X)

ALGEBRAIC_DUAL
    -> defined_without(TOPOLOGY)

# ---------- CONTINUOUS DUAL ----------
if
    X normed_or_topological_vector_space

DUAL_SPACE
    := X_star

X_star
    -> CONTINUOUS_LINEAR_FUNCTIONALS_ON(X)

CONTINUOUS_DUAL
    -> may_be_smaller_than(ALGEBRAIC_DUAL)
       in(INFINITE_DIMENSIONAL_SPACES)

# ---------- BOUNDEDNESS ----------
for normed space X:

f
    -> bounded
       iff
       exists C >= 0
       such_that

abs(f(x))
    <=
C||x||

for_all x

LINEAR_FUNCTIONAL
    -> continuous
       iff
       bounded

# ---------- DUAL NORM ----------
for
    f in X_star

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

equivalently

||f||
    =
sup(
    abs(f(x)) / ||x|| :
    x != 0
)

DUAL_NORM
    -> operator_norm_of(FUNCTIONAL)

# ---------- NORM INEQUALITY ----------
abs(f(x))
    <=
||f|| * ||x||

DUAL_NORM
    -> smallest_bound_constant

# ---------- DUAL SPACE COMPLETENESS ----------
for any normed space X:

X_star
    -> BANACH_SPACE

DUAL_COMPLETENESS
    -> does_not_require(X_COMPLETE)

# ---------- DUAL PAIRING ----------
DUAL_PAIRING
    := <f,x>

where
    f in X_star
    x in X

<f,x>
    := f(x)

DUAL_PAIRING
    -> connects(PRIMAL_SPACE, DUAL_SPACE)

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

if
    x != 0
then

HAHN_BANACH
    -> supports(EXISTENCE_OF_CONTINUOUS_FUNCTIONAL_SEPARATING_x_FROM_ZERO)
       in(NORMED_SPACE)

# ---------- POINT SEPARATION ----------
for
    x != y
    in normed space X

exists
    f in X_star

such_that

f(x) != f(y)

DUAL_SPACE
    -> separates(POINTS)
       via(HAHN_BANACH)

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

every
    f in H_star

-> exists unique y in H

such_that

f(x)
    = <x,y>

subject_to
    INNER_PRODUCT_LINEARITY_CONVENTION

RIESZ_REPRESENTATION
    -> identifies(CONTINUOUS_LINEAR_FUNCTIONALS_WITH_VECTORS)

# ---------- Lp DUALITY ----------
for
    1 < p < infinity

let
    1/p + 1/q = 1

under standard measure-space hypotheses:

(L^p)_star
    -> represented_by(L^q)

functional:

f_g(h)
    =
integral(
    h * conjugate(g)
    dmu
)

LP_DUALITY
    -> uses(HOLDER_INEQUALITY)

# ---------- L1 / L-INFINITY CAUTION ----------
(L^1)_star
    -> often_identified_with(L^infinity)
       under(STANDARD_MEASURE_THEORETIC_CONDITIONS)

(L^infinity)_star
    -/> generally_equal_to(L^1)

DUALITY
    -> depends_on(MEASURE_SPACE_STRUCTURE)

# ---------- WEAK TOPOLOGY ----------
WEAK_TOPOLOGY
    := coarsest_topology_on(X)
       making_all(
           f in X_star
       )
       continuous

LINEAR_FUNCTIONALS
    -> define(WEAK_CONVERGENCE)

x_n
    -> weakly_converges_to(x)

iff

f(x_n)
    -> f(x)

for_all
    f in X_star

# ---------- WEAK STAR TOPOLOGY ----------
on X_star:

WEAK_STAR_TOPOLOGY
    := coarsest_topology
       making(
           f -> f(x)
       )
       continuous
       for_each x in X

FUNCTIONAL_EVALUATION
    -> defines(WEAK_STAR_CONVERGENCE)

# ---------- ANNIHILATOR ----------
for subset M of X:

ANNIHILATOR(M)
    := {
        f in X_star :
        f(m)=0
        for_all m in M
    }

ANNIHILATOR
    -> subspace_of(DUAL_SPACE)

ANNIHILATOR
    -> encodes(LINEAR_CONSTRAINTS)

# ---------- HYPERPLANES ----------
for nonzero continuous linear functional f:

LEVEL_SET
    := {
        x :
        f(x)=c
    }

if
    c = 0
then
    -> KERNEL_HYPERPLANE

if
    c != 0
then
    -> AFFINE_HYPERPLANE

LINEAR_FUNCTIONAL
    -> defines(HYPERPLANES)

# ---------- DISTRIBUTIONS ----------
TEST_FUNCTION_SPACE
    -> topological_vector_space

DISTRIBUTION
    := continuous_linear_functional_on(TEST_FUNCTION_SPACE)

DISTRIBUTION_THEORY
    -> extends(LINEAR_FUNCTIONAL_CONCEPT)

DIRAC_DELTA
    -> distributional_linear_functional

delta_x0(phi)
    := phi(x0)

# ---------- OPTIMISATION ----------
LINEAR_FUNCTIONAL
    -> objective_in(LINEAR_OPTIMISATION)

f(x)
    -> scalar_objective_value

CONSTRAINT_FUNCTIONALS
    -> define(HYPERPLANES_AND_HALFSPACES)

DUALITY
    -> central_to(CONVEX_OPTIMISATION)

# ---------- APPLICATIONS ----------
LINEAR_FUNCTIONAL
    -> applied_to(FUNCTIONAL_ANALYSIS)
    -> applied_to(DUALITY_THEORY)
    -> applied_to(OPTIMISATION)
    -> applied_to(DISTRIBUTION_THEORY)
    -> applied_to(PDEs)
    -> applied_to(QUANTUM_MECHANICS)
    -> applied_to(NUMERICAL_ANALYSIS)

# ---------- RELATION GRAPH ----------
VECTOR_SPACE
    -> domain_of(LINEAR_FUNCTIONAL)

LINEAR_FUNCTIONAL
    -> maps_to(SCALAR_FIELD)

LINEAR_FUNCTIONAL
    -> has(KERNEL)

CONTINUOUS_LINEAR_FUNCTIONAL
    -> element_of(DUAL_SPACE)

DUAL_NORM
    -> measures(FUNCTIONAL_SIZE)

HAHN_BANACH
    -> extends(FUNCTIONALS)

RIESZ_REPRESENTATION
    -> represents(HILBERT_SPACE_FUNCTIONALS_BY_VECTORS)

LINEAR_FUNCTIONALS
    -> generate(WEAK_TOPOLOGY)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "linear functional" ~= "linear scalar-valued map on a vector space",
    "linear form" ~= "linear functional",
    "continuous linear functional" ~= "bounded scalar-valued linear operator on a normed space",
    "dual vector" ~= "element of the dual space in linear and functional analysis",
    "dual pairing" ~= "evaluation of a functional on a vector",
    "kernel of a functional" ~= "subspace of vectors mapped to zero"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    linear_functional_codomain_is_scalar_field: true,
    linearity_requires_additivity_and_homogeneity: true,
    algebraic_dual_and_continuous_dual_are_distinct_concepts: true,
    continuous_linear_functional_is_bounded_on_normed_space: true,
    dual_norm_requires_normed_primal_space: true,
    riesz_representation_requires_hilbert_structure: true,
    lp_duality_requires_measure_theoretic_hypotheses: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    linearity_structure: 0.99,
    kernel_relation: 0.99,
    bounded_continuous_relation: 0.99,
    dual_space_relation: 0.99,
    hahn_banach_riesz_relations: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(vector_space)
    -> identify(candidate_scalar_map)
    -> verify(additivity)
    -> verify(homogeneity)
    -> classify(as_linear_functional_or_not)
    -> determine(kernel_and_range_if_requested)
    -> identify(topological_or_normed_structure)
    -> test(continuity_or_boundedness_if_relevant)
    -> compute(dual_norm_if_available)
    -> apply(hahn_banach_or_riesz_if_conditions_hold)
    -> analyse(weak_topology_or_duality_if_requested)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed linear-functional semantic graph",
    primary_entity: LINEAR_FUNCTIONAL,
    preserve: [
        vector_space,
        scalar_field,
        mapping_formula,
        kernel,
        range,
        topology,
        norm,
        continuity_status
    ],
    infer_through: [
        algebraic_duals,
        continuous_duals,
        boundedness,
        dual_norms,
        hahn_banach,
        riesz_representation,
        weak_topologies,
        annihilators
    ],
    avoid: [
        treating_vector_valued_operator_as functional,
        assuming_all_algebraic_functionals_are continuous,
        confusing_algebraic_dual_with continuous dual,
        applying_riesz_without Hilbert structure,
        asserting_Lp_duality_without measure conditions,
        ignoring_complex_conjugation conventions
    ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + LINEAR_FUNCTIONAL
    + KERNEL
    + DUAL_SPACE
    + DUAL_NORM
    + HAHN_BANACH_THEOREM
    + RIESZ_REPRESENTATION
    + WEAK_TOPOLOGY
    + ANNIHILATOR
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> LINEAR_FUNCTIONAL
    -> VECTOR_SPACE_IDENTIFICATION
    -> SCALAR_MAP_IDENTIFICATION
    -> LINEARITY_VALIDATION
    -> KERNEL_RANGE_ANALYSIS
    -> TOPOLOGICAL_NORMED_STRUCTURE_ANALYSIS
    -> CONTINUITY_BOUNDEDNESS_ANALYSIS
    -> DUAL_NORM_ANALYSIS
    -> DUALITY_THEOREM_ANALYSIS
    -> WEAK_TOPOLOGY_ANALYSIS
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW