Cauchy-Schwarz Inequality

Equivalent forms

\left|\int f^*g\,dx\right|^2\le \int|f|^2dx\int|g|^2dx

A one-line inequality that silently powers the triangle inequality, correlation bounds, and the derivation of the uncertainty principle.

Symbol	Name	SI unit	Dimension	Range
$\langle u,v\rangle$	Inner productoutput Overlap of the two vectors	1	1	—
$\lVert u\rVert$	Norm of u Length of the first vector	1	1	0.1 – 5
$\lVert v\rVert$	Norm of v Length of the second vector	1	1	0.1 – 5
$\theta$	Angle Angle between the vectors	1	1	0 – 3.14

Where it holds

Holds in any real or complex inner-product space (finite or infinite dimensional), including function spaces and the space of quantum states.

Dimensional analysis

|⟨u,v⟩| carries dimension of u times v

‖u‖‖v‖ same product

Both sides scale identically; for the dimensionless vectors here they are pure numbers.

\checkmark

Discovery

Cauchy, Bunyakovsky, Schwarz · 1821

Cauchy proved the sum version in 1821; Bunyakovsky extended it to integrals in 1859; Schwarz gave the modern inner-product-space form in 1888. The triple name honors all three, though usage varies by country — 'Cauchy–Schwarz' in the West, 'Cauchy–Bunyakovsky–Schwarz' in Russia.

Research status: stable

Real-world applications

Bounding statistical correlation coefficients to $[-1$ ,1]
Deriving the Heisenberg uncertainty relation
Convergence proofs and error bounds in numerical analysis

Common misconceptions

Equality holds only for parallel (linearly dependent) vectors, not merely orthogonal ones
It applies to functions and quantum states, not just arrows in space
The complex form needs the modulus |⟨u,v⟩|, since the inner product can be complex

Experimental verification

A theorem, not measured; its consequences (|correlation|

\le 1

,

\Delta x\Delta p \ge \hbar /2)

are confirmed experimentally everywhere they apply.

Derivation

For any scalar

\lambda

,

0 \le

‖

u - \lambda v

‖

^{2}

= ‖u‖

^{2} - 2Re(\lambda

*⟨u,v⟩

) + |\lambda |^{2}

‖v‖

^{2}

.

Choose

\lambda

= ⟨u,v⟩/‖v‖

^{2} to

minimize the right side; substituting gives

0 \le

‖u‖

^{2} -

|⟨u,v⟩|

^{2}

/‖v‖

^{2}

, i.e. |⟨u,v⟩|

^{2}

≤ ‖u‖

^{2}

‖v‖

^{2}

.

Equality requires

u - \lambda v = 0

, i.e. the vectors are linearly dependent.

What if…

What if u and v are parallel?

$\cos \theta = \pm 1$ and the inequality becomes an equality — the overlap saturates its maximum.

What if they are orthogonal?

⟨u,v⟩ $= 0$ , the weakest possible overlap — the inequality is satisfied with a huge margin.

1

Uncertainty principle step

Given · states |

\Delta A \psi

⟩ and |

\Delta B \psi

⟩

Find · lower bound

on \Delta A\cdot \Delta B

Steps

Apply |⟨f,g⟩| ≤ ‖f‖‖g‖ with $f=\Delta$ Â $\psi$ , $g=\Delta B$ ̂ $\psi$
‖f‖‖g‖ $= \Delta A\cdot \Delta B$ , |⟨f,g⟩| $\ge |Im$ ⟨f,g⟩| = |⟨[Â,B̂]⟩|/2
$\Delta A\cdot \Delta B$ ≥ |⟨[Â,B̂]⟩|/2

Result · Cauchy–Schwarz gives

\Delta A\cdot \Delta B

≥ |⟨[Â,B̂]⟩|/2

triangle inequalityHeisenberg Uncertainty Principle→inner product