### Inner Product Spaces: Conclusion of Inner product spaces

### End of Inner product spaces

The inequality of Cauchy-Schwarz is named after A.-L. Cauchy (1789--1857) and H.A. Schwarz (1843--1921). The former described the inequality in terms of sequences of numbers, Schwarz in terms of function spaces. Independent of these two people, the inequality is also derived by V.Y. Bunyakovsky (1804--1889) also in the context of function spaces.

Inner product spaces of functions is dealt with in more detail in advanced analysis units; applications of these types of inner product spaces are found in signal analysis and in quantum mechanics.

A variant of the inner product in which vectors don't have a positive length per se, is found in the theory of relativity. Implicitly this is already dealt with when classifying quadratic forms from Linear Algebra 2.

The Pythagorean theorem is of course coming from geometry, but if applied in conveniently choses inner product spaces of functions, also has surprising consequences important in analysis. An example of this is the inequality \[

\sum_{n=1}^{\infty} \frac{1}{n^2}\leq \frac{\pi^2}{6}

\]that can be derived with the help of an inner product space.

The procedure of Gram-Schmidt is named after the danishman J.P. Gram (1850--1916) and the german E. Schmidt (1876--1959) and was introduced in the context of vectors spaces of functions.

Perpendicular projections can be used to derive the least squares method; this method is for example an essential ingredient of physics practice. The notions of length, angle and perpendicular will return in Linear Algebra 2 when studying orthagonal images, like reflections and rotations.

And although we did not go into it, matrices can be used to define the data of an inner product, in the form of the so called Gram-matrix. If #\basis{\vec{u}_1, \ldots ,\vec{u}_n}# is a bas of an inner product space #V#, then the #(i,j)#-element of this #(n\times n)#-matrix #\dotprod{\vec{u}_i}{\vec{u}_j}#.

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