Cauchy-Schwarz Formula

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One of the most important inequalities in mathematics is inarguably the famous Cauchy-Schwarz inequality whose use appears in many important proofs. We will prove this important inequality and prove an analogue of the triangle inequality in higher dimension Euclidean -space.

The Cauchy-Schwarz Inequality

Theorem 1 (The Cauchy-Schwarz Inequality): If then .

  • Proof: Let . Then we want to prove that:

  • Notice that the sum of squares is always nonnegative, and so for all we have that:

  • Let , , and . Then:

  • Suppose that . Then reduces to which is true. If , then let . Then:

  • Therefore we have that:

Often times the Cauchy-Schwarz inequality is stated by squaring both sides of the inequality above:

The Triangle Inequality

Theorem 2 (The Triangle Inequality): If then .

  • Proof: Let . Then:

  • Square both sides of the equation and apply the Cauchy-Schwarz inequality at to get:

  • Square rooting both sides of the inequality above yields as desired.


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