Alphabeta MathHow 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.
FALSE: makes an inner product space
Statement
False claim: the standard form on is an inner product.
Facts & Assumptions
Given: the vector .
Over , one has (The standard bilinear form on ).
Refutation
The vector is nonzero, but [L1] gives in .
An inner product cannot vanish on a nonzero vector, so the false claim fails.
Remarks
- What remains true is that the form is symmetric and bilinear. That is all the page ever uses over .
Depends on
- The standard bilinear form $\langle x,y\rangle=\sum_{i<n}x_iy_i$ on $F^{n}$
- Real and complex inner product spaces, with the inner product linear in the first argument
- Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms
- For every prime $p$, the two operations on $\mathbb{Z}/p$ make it a field
- Diagonal criterion: if $\langle v_i,w_i\rangle\ne0$ and $\langle v_i,w_j\rangle=0$ for $i\ne j$, then $v_1,\dots,v_m$ are linearly independent
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
19 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
- L. Babai and P. Frankl, Linear Algebra Methods in Combinatorics, §2.3.1 (standard reference, not scraped)