ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-13

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.

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Coordinate vectors converge weakly to zero in ell p

Example

In real or complex $ℓ^{p}$ , $1 < p < \infty$ , the coordinate vectors $e_{n}$ converge weakly to zero, although $∥ e_{n} ∥_{p} = 1$ .

Facts & Assumptions

[F1]

Put $q = p / (p - 1)$ , so $1 / p + 1 / q = 1$ (Conjugate exponents, including the endpoint conventions). The sequence norm is $∥ x ∥_{p} = (\sum_{k} ∣ x_{k} ∣^{p})^{1 / p}$ ( $ℓ^{p}$ is the $L^{p}$ space of counting measure); for complex sequences apply this to their real nonnegative moduli.

[F2]

Positive real powers obey exponent laws (The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents), and finite Hölder applies to nonnegative real moduli (Holder's inequality for finite sums and conjugate real exponents). For a nonzero complex $b$ , $b \overline{b} = ∣ b ∣^{2}$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[F3]

Weak convergence tests every bounded scalar-linear functional (Weak convergence of nets and sequences).

Verification

Given: $1 < p < \infty$ and a bounded scalar-linear $F$ on $ℓ^{p}$ .

1.1

Put $b_{k} = F (e_{k})$ . For a finite initial segment let $v_{k} = {\overline{b}}_{k} ∣ b_{k} ∣^{q - 2}$ if $b_{k} \neq 0$ , and $v_{k} = 0$ if $b_{k} = 0$ ; set other coordinates zero. For real scalars conjugation fixes $b_{k}$ . Then $b_{k} v_{k} = ∣ b_{k} ∣^{q}$ and $∣ v_{k} ∣^{p} = ∣ b_{k} ∣^{(q - 1) p} = ∣ b_{k} ∣^{q}$ . If $A_{N} = \sum_{k < N} ∣ b_{k} ∣^{q}$ , linearity and boundedness give $A_{N} = F (v) \leq ∥ F ∥ A_{N}^{1 / p}$ . If $A_{N} = 0$ there is nothing to divide; otherwise $A_{N}^{1 / q} \leq ∥ F ∥$ .

givenF1F2algebra

2.1

The nondecreasing partial sums $A_{N}$ are bounded by $∥ F ∥^{q}$ , so their nonnegative series converges and $b_{k} \to 0$ . Indeed infinitely many $∣ b_{k} ∣ \geq ε > 0$ would make arbitrarily large finite sums exceed that bound. For completeness finite Hölder bounds $\sum_{k < N} ∣ x_{k} b_{k} ∣ \leq ∥ x ∥_{p} ∥ b ∥_{q}$ , so the coefficient pairing is absolutely convergent, consistently over both fields. In particular $F (e_{n}) = b_{n} \to 0$ for every $F$ , giving weak convergence. Directly $∥ e_{n} ∥_{p} = (1^{p})^{1 / p} = 1$ , so there is no norm convergence to zero. $□$

step 1.1F1F2F3

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Bühler–Salamon, Functional Analysis (2017); exact harvest in batch coverage (standard reference, not scraped)
Teschl, Topics in Real and Functional Analysis (2017); exact harvest in batch coverage (standard reference, not scraped)