DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Complex norm and complex normed space

Definition

Let $V$ be a vector space over $C$ (Vector space over a field, with the complex field of The complex numbers as $R [x] / (x^{2} + 1)$ , with the real embedding and imaginary unit $i$ ). A complex norm on $V$ is a function $N : V \to R$ such that, for all $u, v \in V$ and every $λ \in C$ :

$N (v) = 0$ if and only if $v = 0$ ;
$N (λ v) = ∣ λ ∣_{C} N (v)$ , where $∣ λ ∣_{C}$ is the complex modulus of Real and imaginary parts, complex conjugation, and modulus;
$N (u + v) \leq N (u) + N (v)$ .

A complex normed space is a pair $(V, N)$ with $V$ a complex vector space and $N$ a complex norm on $V$ . Since the codomain is $R$ , every norm value is a finite real number; extended-real values are not allowed. The homogeneity clause is for every complex scalar, not merely real scalars.

Depends on

Used by

A finite-dimensional normed subspace is closed Corollary
The induced length is a norm Corollary
Weak differentiation has a closed graph on its natural domains Corollary
A complete domain is necessary for sequential uniform boundedness Counterexample
A first resolvent estimate does not ensure the prescribed semigroup bound Counterexample
A bounded bilinear map between normed spaces Definition
A bounded linear operator between normed spaces Definition
A complex normed linear subspace Definition
Absorbing, balanced, and absolutely convex sets Definition
Banach space Definition
C star algebra Definition
Compact linear operator Definition
Compactly embedded normed spaces Definition
Complex sector and bounded analytic semigroup Definition
Convex sets and continuous real-hyperplane separation in a normed space Definition
Equivalent complex norms Definition
Resolvent and spectrum of a closed operator on a Banach space Definition
Sectorial operator with the semigroup sign convention Definition
Strongly continuous semigroup Definition
The dual space X^* of a normed space and its dual norm Definition
A compact group with no faithful continuous finite-dimensional representation Example
A real norm on C that is not a complex norm Example
An explicit norming functional for the finite-dimensional maximum norm Example
Coordinate partial sums on c₀ Example
Fredholm alternative for an integral equation Example
The complex line through (1,i) has a two-vector real basis Example
A compact set and a disjoint closed set have a positive norm-distance gap Lemma
A complex normed subspace and its finite basis realify Lemma
Banach-valued power series are determined by their real values Lemma
Canonical Banach complexification of a real Banach space Lemma
Kernel of identity minus compact is finite dimensional Lemma
Realification preserves a complex norm and its metric Lemma
Resolvent identity and holomorphy for a closed operator Lemma
The complex coordinate sum norm, including the zero space Lemma
Vector addition and scalar multiplication are continuous in a normed space Lemma
A bounded complex linear functional on a subspace of a complex normed space extends with the same norm Theorem
A bounded linear functional on an arbitrary subspace extends with the same norm, without assuming the subspace is closed Theorem
A chosen algebraic basis identifies a finite-dimensional normed space with a coordinate space Theorem
Jordan–von Neumann: a norm is induced by an inner product exactly when it satisfies the parallelogram law Theorem
Meromorphic continuation and intertwining identity for A(nu) Theorem

…and 1 more result.

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Tomasz Kochanek, Functional analysis, Lecture 1 (complete five-page lecture notes) (standard reference, not scraped)
M. Winklmeier, Functional Analysis lecture notes (complete 140-page lecture notes) (standard reference, not scraped)