Five variants of the Cauchy–Schwarz inequality | Tangente

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The Cauchy–Schwarz inequality follows from the fact that, in a Euclidean plane (that is, one equipped with an inner product), the equality $\mathrm{\mid }(\overrightarrow{u},\overrightarrow{v})\mathrm{\mid }\le \mathrm{\parallel }\overrightarrow{u}\mathrm{\parallel }\mathrm{\parallel }\overrightarrow{v}\mathrm{\parallel }|\; \backslash left(\; \backslash overrightarrow\{u\},\; \backslash overrightarrow\{v\}\; \backslash right)|\; \backslash leq\; \backslash Vert\; \backslash overrightarrow\{u\}\; \backslash Vert\; \backslash Vert\; \backslash overrightarrow\{v\}\; \backslash Vert$ holds for every pair of nonzero vectors. This inequality expresses the fact that the absolute value of a cosine is less than or equal to 1. For n = 2, however, the inequality can be proved by a simple calculation or by a geometric argument.

Geometric inequality: the area of a parallelogram is less than that of a rectangle with corresponding side lengths.

Arithmetic inequality, or Cauchy's inequality for n-tuples: let ( a₁, a₂, a₃… *a_n ) and ( b*₁, b₂, b₃… *b_n ) be two n-tuples* of positive real numbers.

Then: