TheoremStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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.

Holder's inequality for finite sums and conjugate real exponents

Statement

Let $p,q>1$ with $1/p+1/q=1$ . For real families $(a_i)_{i<n}$ and $(b_i)_{i<n}$ , $\sum_{i<n}|a_ib_i|\le\left(\sum_{i<n}|a_i|^p\right)^{1/p}\left(\sum_{i<n}|b_i|^q\right)^{1/q}.$

Facts & Assumptions

Given: A natural $n$ , conjugate exponents $p,q>1$ , and real families $a_i,b_i$ for $i<n$ .

[L1]

Young's inequality says $uv\le u^p/p+v^q/q$ for $u,v\ge0$ (Young's inequality for conjugate real exponents).

[L2]

Finite sums obey termwise addition and scalar multiplication, including the empty-sum convention, and $|ab|=|a||b|$ (Finite sums and finite products, by recursion, Laws of finite sums and finite products, Basic properties of the absolute value).

[L3]

For $r>0$ , the zero-base convention gives $0^r=0$ , while positive-base real powers are positive and obey the real-power laws (Real powers for positive bases, with the zero-base positive-exponent convention, The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents, The exponential is positive and satisfies $\exp(-x)=1/\exp(x)$ ).

Proof

technique · direct

1.1

Put $A=(\sum_{i<n}|a_i|^p)^{1/p}$ and $B=(\sum_{i<n}|b_i|^q)^{1/q}$ . If $A=0$ or $B=0$ , the real-power laws and zero convention show that the corresponding nonnegative power sum is zero; finite-sum order then makes every corresponding term zero, so the claim follows.

L2L3

1.2

Suppose $A,B>0$ , and set $u_i=|a_i|/A$ , $v_i=|b_i|/B$ . Then $\sum u_i^p=\sum v_i^q=1$ .

L2L3

2.1

Apply [L1] to $u_i,v_i$ and sum over $i<n$ to obtain $\sum u_iv_i\le1/p+1/q=1$ .

step 1.2L1L2

3.1

Multiplying by $AB$ and using $|a_ib_i|=|a_i||b_i|$ gives the asserted inequality.

step 1.2step 2.1L2∎

Depends on

Used by

Minkowski's inequality for finite sums and real exponent p greater than one Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 51 results over 21 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)
MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)