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PathsCauchy-Schwarz inner-product bound

Theorem trail

Cauchy-Schwarz inner-product bound

The geometry step behind covariance, projection, and information bounds.

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Research Record

A compact audit view for the theorem before the full source, proof, diagnostic, and failure-mode sections.

Claim scope
theorem
Evidence level
Exact Lean wrapper
Source precision
reviewed / B
Diagnostics
1 item
Exercises
1 exercise
Failure checks
2 checks
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Theorem Statement

The inner product of two vectors cannot exceed the product of their lengths. This is a generalization of the fact that .

Assumptions

  • vectors in an inner product space

Proof Sketch

For nonzero v, minimize the squared norm of u - tv as a function of the scalar t. The resulting quadratic is always nonnegative, so its discriminant is nonpositive. Rearranging gives |<u,v>| <= ||u|| ||v||. The zero-vector cases are immediate.

Proof Obligations

Inner-product norm and zero-vector cases

assumption

Verify the norm is induced by the same inner product and handle the zero-vector cases before using the quadratic argument.

Mixing norms or skipping zero cases can turn a standard inequality into an invalid algebraic shortcut.

Quadratic nonnegativity gives the discriminant bound

proof step

The proof minimizes ||u - tv||^2 or uses its nonnegative discriminant to force |<u,v>| <= ||u|| ||v||.

Quoting the inequality without the quadratic step hides the equality condition and projection geometry.

Covariance uses need their own setup

boundary

Probabilistic covariance and information-inequality uses require the variables to live in the right L2 space before Cauchy-Schwarz applies.

Moving from finite-dimensional vectors to random variables without integrability checks overextends the theorem.

Norm and inner product must match

boundary

Use the inequality with the norm induced by the same inner product; unrelated norms need their own comparison theorem.

Mixing geometric structures can make a true inner-product theorem look like an unsupported norm inequality.

Source Locators

Linear Algebra Done Right (2024)

proof

Chapter 6 — Inner Product Spaces

axler-2024:ch6

kind
chapter
review
reviewed / B
extract
manual

Chapter-level support for the real inner-product-space Cauchy-Schwarz inequality.

Linear Algebra Done Right (2024)

proof

6.14 — Cauchy-Schwarz inequality

axler-2024:thm6.14

kind
theorem
review
reviewed / A
extract
manual
number
6.14

Anchor: Cauchy-Schwarz inequality

Reviewed theorem-level support for the Cauchy-Schwarz inequality in inner product spaces.

Curated Trace

  1. 1
    Inner product geometrysetting

    Lengths, angles, and projections are encoded by the inner product.

    Checkpoint

    Confirm the norm is the one induced by the inner product.

  2. 2
    Cauchy-Schwarztheorem

    The quadratic nonnegativity argument bounds any inner product by two norms.

    Checkpoint

    Equality is a collinearity condition, not a generic case.

  3. 3
    Score covariance boundstatistics use

    Information inequalities often bound a covariance by L2 norms.

    Checkpoint

    Identify the estimator term and the score term before applying the theorem.

  4. 4
    Cramer-Rao stepdownstream

    The lower bound follows after Cauchy-Schwarz controls the score-estimator covariance.

    Checkpoint

    Unbiasedness and regularity must be checked before the rearrangement.

Failure Modes

Covariance uses need their own setup

Probabilistic covariance and information-inequality uses require the variables to live in the right L2 space before Cauchy-Schwarz applies.

Risk: Moving from finite-dimensional vectors to random variables without integrability checks overextends the theorem.

Check: Probabilistic covariance and information-inequality uses require the variables to live in the right L2 space before Cauchy-Schwarz applies.

Norm and inner product must match

Use the inequality with the norm induced by the same inner product; unrelated norms need their own comparison theorem.

Risk: Mixing geometric structures can make a true inner-product theorem look like an unsupported norm inequality.

Check: Use the inequality with the norm induced by the same inner product; unrelated norms need their own comparison theorem.