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

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

Statement

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

Facts & Assumptions

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

[L1]

Holder's inequality holds for finite sums and conjugate real exponents (Holder's inequality for finite sums and conjugate real exponents).

[L2]

Finite sums distribute over addition, and $|u+v|\le|u|+|v|$ (Finite sums and finite products, by recursion, Laws of finite sums and finite products, Basic properties of the absolute value).

[L3]

Positive-base real-power laws hold; the zero-base positive-exponent convention gives $0^r=0$ , and positive-base real powers are positive (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

Let $q=p/(p-1)$ and put $C=(\sum_{i<n}|a_i+b_i|^p)^{1/p}$ . If $C=0$ , the claim is immediate.

L2L3

1.2

For $C>0$ , multiply $|a_i+b_i|^{p-1}$ by $|a_i+b_i|\le|a_i|+|b_i|$ and sum to get $C^p\le\sum|a_i||a_i+b_i|^{p-1}+\sum|b_i||a_i+b_i|^{p-1}$ .

L2L3

2.1

Apply Holder to both sums in step 1.2. Since $(p-1)q=p$ , their common second factor is $C^{p-1}$ .

step 1.2L1L3

3.1

Thus $C^p\le(A+B)C^{p-1}$ , where $A,B$ are the two right-side norms; dividing by $C^{p-1}>0$ proves the claim.

step 1.1step 2.1L3∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 51 results over 22 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)