Norm Ontology

flow "NormOntology" v1.0

# ============================================================
# NORM ONTOLOGY — FLOW ALGEBRA
# ============================================================

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

# ---------- CORE ENTITY ----------
NORM := {
    type: MathematicalFunction,
    domain: FunctionalAnalysis,
    purpose: "Assign a nonnegative magnitude or size to vectors while satisfying definiteness, absolute homogeneity and the triangle inequality."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    NORM,
    VECTOR_SPACE,
    NORMED_SPACE,
    MAGNITUDE,
    DISTANCE,
    METRIC,
    UNIT_BALL,
    UNIT_SPHERE,
    CONVERGENCE,
    CAUCHY_SEQUENCE,
    COMPLETENESS,
    BANACH_SPACE,
    INNER_PRODUCT,
    EQUIVALENT_NORMS,
    OPERATOR_NORM
}

# ---------- BASIC DEFINITION ----------
let
    V
        := VECTOR_SPACE
        over(SCALAR_FIELD)

NORM
    := mapping(
        V -> NONNEGATIVE_REAL_NUMBERS
    )

x
    -> ||x||

# ---------- NORM AXIOMS ----------
for all
    x,y in V
    and scalar alpha

NONNEGATIVITY:
    ||x|| >= 0

DEFINITENESS:
    ||x|| = 0
    iff
    x = ZERO_VECTOR

ABSOLUTE_HOMOGENEITY:
    ||alpha*x||
    =
    abs(alpha)*||x||

TRIANGLE_INEQUALITY:
    ||x+y||
    <=
    ||x|| + ||y||

NORM
    -> must_satisfy(ALL_NORM_AXIOMS)

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

NORM
    -> equips(VECTOR_SPACE_WITH_SIZE_STRUCTURE)

NORMED_SPACE
    -> supports(DISTANCE)
    -> supports(CONVERGENCE)
    -> supports(CAUCHY_SEQUENCES)

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

NORM
    -> induces(METRIC)

INDUCED_METRIC
    -> translation_invariant

d(x+a,y+a)
    = d(x,y)

# ---------- REVERSE TRIANGLE INEQUALITY ----------
abs(
    ||x|| - ||y||
)
    <=
||x-y||

REVERSE_TRIANGLE_INEQUALITY
    -> implies(NORM_FUNCTION_IS_CONTINUOUS)

# ---------- UNIT BALL ----------
OPEN_UNIT_BALL
    := {
        x :
        ||x|| < 1
    }

CLOSED_UNIT_BALL
    := {
        x :
        ||x|| <= 1
    }

UNIT_SPHERE
    := {
        x :
        ||x|| = 1
    }

NORM
    -> defines(GEOMETRIC_BALLS_AND_SPHERES)

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

iff

||x_n-x||
    -> 0

NORM_CONVERGENCE
    -> convergence_in(INDUCED_METRIC)

# ---------- CAUCHY SEQUENCE ----------
x_n
    -> norm_Cauchy

iff

for_every epsilon > 0
exists N
such_that

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

CAUCHY_SEQUENCE
    -> convergence_candidate

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

iff

every_norm_Cauchy_sequence
    -> converges_in_space

COMPLETE_NORMED_SPACE
    ~= BANACH_SPACE

# ---------- EUCLIDEAN NORM ----------
for
    x = (x_1,...,x_n)

L2_NORM:
    ||x||_2
    =
    sqrt(
        sum(
            abs(x_i)^2
        )
    )

EUCLIDEAN_NORM
    -> induced_by(STANDARD_INNER_PRODUCT)

# ---------- L1 NORM ----------
||x||_1
    =
sum(
    abs(x_i)
)

L1_NORM
    -> measures(SUM_OF_COMPONENT_MAGNITUDES)

# ---------- L-INFINITY NORM ----------
||x||_infinity
    =
max_i(
    abs(x_i)
)

LINFINITY_NORM
    -> measures(MAXIMUM_COMPONENT_MAGNITUDE)

# ---------- LP NORMS ----------
for
    1 <= p < infinity

||x||_p
    =
(
    sum(
        abs(x_i)^p
    )
)^(1/p)

LP_NORM
    -> satisfies(TRIANGLE_INEQUALITY)
       via(MINKOWSKI_INEQUALITY)

# ---------- FUNCTION NORMS ----------
for measurable function f:

Lp_NORM
    :=
(
    integral(
        abs(f)^p
        dmu
    )
)^(1/p)

for
    1 <= p < infinity

SUPREMUM_NORM
    := sup_x(abs(f(x)))

ESSENTIAL_SUPREMUM_NORM
    := ess_sup(abs(f))

FUNCTION_NORM
    -> measures(FUNCTION_SIZE)

# ---------- INNER PRODUCT NORM ----------
if
    V has_inner_product
then

||x||
    := sqrt(<x,x>)

INNER_PRODUCT
    -> induces(NORM)

INNER_PRODUCT_NORM
    -> satisfies(PARALLELOGRAM_IDENTITY)

# ---------- PARALLELOGRAM LAW ----------
if
    norm arises_from_inner_product
then

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

PARALLELOGRAM_IDENTITY
    -> characterises(INNER_PRODUCT_NORMS)
       via(JORDAN_VON_NEUMANN_THEOREM)

# ---------- EQUIVALENT NORMS ----------
NORM_1
    and NORM_2
    -> equivalent

if
    exists c,C > 0
    such_that

c||x||_1
    <=
||x||_2
    <=
C||x||_1

for_all x

EQUIVALENT_NORMS
    -> generate(SAME_TOPOLOGY)

# ---------- FINITE-DIMENSIONAL EQUIVALENCE ----------
on finite-dimensional vector space:

ALL_NORMS
    -> equivalent

FINITE_DIMENSION
    -> topology_independent_of(NORM_CHOICE)

IN_INFINITE_DIMENSIONS
    -> norms_need_not_be_equivalent

# ---------- OPERATOR NORM ----------
for
    T: X -> Y
    bounded_linear

||T||
    :=
sup(
    ||T(x)||_Y :
    ||x||_X <= 1
)

OPERATOR_NORM
    -> measures(MAXIMUM_AMPLIFICATION)

||T(x)||_Y
    <=
||T|| * ||x||_X

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

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

DUAL_NORM
    -> operator_norm_of(LINEAR_FUNCTIONAL)

# ---------- MATRIX NORMS ----------
MATRIX_NORM
    -> may_be(INDUCED_OPERATOR_NORM)
    -> may_be(FROBENIUS_NORM)
    -> may_be(OTHER_SUBMULTIPLICATIVE_NORM)

INDUCED_MATRIX_NORM
    := sup(
        ||Ax|| / ||x||
        for x != 0
    )

SUBMULTIPLICATIVITY:
    ||AB||
    <=
    ||A||*||B||

for compatible submultiplicative norms

# ---------- SEMINORM ----------
SEMINORM
    -> satisfies(
        NONNEGATIVITY,
        ABSOLUTE_HOMOGENEITY,
        TRIANGLE_INEQUALITY
    )

SEMINORM
    -/> requires(
        ||x|| = 0 implies x = 0
    )

NORM
    -> definite(SEMINORM)

# ---------- QUOTIENT NORM ----------
if
    M
        -> closed_linear_subspace_of(X)

for coset
    x + M

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

QUOTIENT_NORM
    -> equips(X/M_WITH_NORM)

# ---------- APPLICATIONS ----------
NORM
    -> applied_to(FUNCTIONAL_ANALYSIS)
    -> applied_to(NUMERICAL_ANALYSIS)
    -> applied_to(OPTIMISATION)
    -> applied_to(APPROXIMATION_THEORY)
    -> applied_to(MACHINE_LEARNING)
    -> applied_to(PDEs)
    -> applied_to(ERROR_ANALYSIS)

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

NORM
    -> induces(METRIC)

NORM
    -> defines(CONVERGENCE)

NORM
    -> defines(CAUCHY_SEQUENCE)

NORMED_SPACE
    + COMPLETENESS
    -> BANACH_SPACE

INNER_PRODUCT
    -> induces(NORM)

BOUNDED_LINEAR_OPERATOR
    -> has(OPERATOR_NORM)

DUAL_SPACE
    -> has(DUAL_NORM)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "norm" ~= "function assigning vector magnitude while satisfying norm axioms",
    "vector norm" ~= "norm on a vector space",
    "induced metric" ~= "distance d(x,y)=||x-y|| generated by a norm",
    "Banach norm" ~= "norm under which the vector space is complete",
    "operator norm" ~= "maximum bounded linear amplification measured relative to domain and codomain norms",
    "equivalent norms" ~= "norms bounded above and below by positive constant multiples of each other"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    norm_must_be_definite: true,
    norm_must_be_absolutely_homogeneous: true,
    norm_must_satisfy_triangle_inequality: true,
    norm_induces_metric_but_not_every_metric_is_norm_induced: true,
    equivalent_norms_generate_same_topology: true,
    all_norms_equivalent_only_in_finite_dimensions_in_general: true,
    operator_norm_depends_on_domain_and_codomain_norms: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    norm_axioms: 0.99,
    metric_relation: 0.99,
    completeness_relation: 0.99,
    lp_norm_relation: 0.99,
    operator_norm_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(vector_space_or_function_space)
    -> identify(candidate_norm)
    -> test(nonnegativity)
    -> test(definiteness)
    -> test(absolute_homogeneity)
    -> test(triangle_inequality)
    -> construct(induced_metric)
    -> analyse(convergence_and_cauchy_structure_if_requested)
    -> test(completeness_if_required)
    -> compare_with(other_norms_if_relevant)
    -> construct(operator_or_dual_norm_if_requested)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed norm semantic graph",
    primary_entity: NORM,
    preserve: [
        vector_space,
        scalar_field,
        norm_formula,
        domain,
        induced_metric,
        convergence_structure,
        completeness_status
    ],
    infer_through: [
        normed_spaces,
        banach_spaces,
        inner_product_norms,
        lp_norms,
        equivalent_norms,
        operator_norms,
        dual_norms
    ],
    avoid: [
        accepting_seminorm_as norm_without definiteness,
        assuming_every_metric_comes_from norm,
        assuming_all_infinite_dimensional_norms_are equivalent,
        confusing_norm_with inner product,
        ignoring_domain_and_codomain norms in operator norm
    ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + NORM
    + NORMED_SPACE
    + METRIC
    + CAUCHY_SEQUENCE
    + COMPLETENESS
    + EQUIVALENT_NORMS
    + OPERATOR_NORM
    + DUAL_NORM
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> NORM
    -> SPACE_IDENTIFICATION
    -> CANDIDATE_NORM_IDENTIFICATION
    -> NORM_AXIOM_VALIDATION
    -> METRIC_CONSTRUCTION
    -> CONVERGENCE_ANALYSIS
    -> COMPLETENESS_ANALYSIS
    -> NORM_EQUIVALENCE_ANALYSIS
    -> OPERATOR_DUAL_NORM_ANALYSIS
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW