Cauchy–Schwarz inequality

Cauchy-Schwarz inequation in Euclidean plane

Cauchy–Schwarz Inequality is a fundamental theorem in the field of mathematics, particularly in linear algebra and analysis. It establishes an important relationship between the dot product of two vectors and the product of their Euclidean norms. The inequality is named after the mathematicians Augustin-Louis Cauchy and Hermann Schwarz, who formulated and proved the inequality in the 19th century.

Statement of the Inequality

The Cauchy–Schwarz Inequality states that for all vectors \(\mathbf{u}\) and \(\mathbf{v}\) in an inner product space, the following inequality holds:

\[ |\langle \mathbf{u}, \mathbf{v} \rangle| \leq \|\mathbf{u}\| \cdot \|\mathbf{v}\| \]

where \(\langle \mathbf{u}, \mathbf{v} \rangle\) denotes the dot product (or inner product) of \(\mathbf{u}\) and \(\mathbf{v}\), and \(\|\mathbf{u}\|\) and \(\|\mathbf{v}\|\) denote the Euclidean norm (or magnitude) of \(\mathbf{u}\) and \(\mathbf{v}\), respectively.

Applications

The Cauchy–Schwarz Inequality has numerous applications across various domains of mathematics and physics. It is used in the proofs of other mathematical theorems, in the study of Fourier series, in quantum mechanics where it has implications for the uncertainty principle, and in statistics for proving the correlation coefficient's properties.

Proof

There are several proofs of the Cauchy–Schwarz Inequality, varying in complexity and mathematical prerequisites. A common approach involves considering the non-negativity of a certain quadratic expression derived from the vectors involved and applying the properties of inner product spaces.

Generalizations

The Cauchy–Schwarz Inequality can be generalized to more abstract settings, such as to any inner product space, including infinite-dimensional spaces. This generalization is crucial in the study of functional analysis and Hilbert spaces.

See Also

• Bessel's Inequality
• Triangle Inequality
• Holder's Inequality
• Minkowski Inequality