**History**

Vector spaces stem from affine geometry, via the introduction of coordinates in the plane or three-dimensional space. Around 1636, Descartes and Fermat founded analytic geometry by equating solutions to an equation of two variables with points on a plane curve. To achieve geometric solutions without using coordinates, Bolzano introduced, in 1804, certain operations on points, lines and planes, which are predecessors of vectors. This work was made use of in the conception of barycentric coordinates by Möbius in 1827. The foundation of the definition of vectors was Bellavitis' notion of the bipoint, an oriented segment one of whose ends is the origin and the other one a target. Vectors were reconsidered with the presentation of complex numbers by Argand and Hamilton and the inception of quaternions and biquaternions by the latter. They are elements in **R**2, **R**4, and **R**8; treating them using linear combinations goes back to Laguerre in 1867, who also defined systems of linear equations.

In 1857, Cayley introduced the matrix notation which allows for a harmonization and simplification of linear maps. Around the same time, Grassmann studied the barycentric calculus initiated by Möbius. He envisaged sets of abstract objects endowed with operations. In his work, the concepts of linear independence and dimension, as well as scalar products are present. Actually Grassmann's 1844 work exceeds the framework of vector spaces, since his considering multiplication, too, led him to what are today called algebras. Peano was the first to give the modern definition of vector spaces and linear maps in 1888.

An important development of vector spaces is due to the construction of function spaces by Lebesgue. This was later formalized by Banach and Hilbert, around 1920. At that time, algebra and the new field of functional analysis began to interact, notably with key concepts such as spaces of *p*-integrable functions and Hilbert spaces. Vector spaces, including infinite-dimensional ones, then became a firmly established notion, and many mathematical branches started making use of this concept.

Read more about this topic: Vector Space

### Famous quotes containing the word history:

“The *history* of mankind interests us only as it exhibits a steady gain of truth and right, in the incessant conflict which it records between the material and the moral nature.”

—Ralph Waldo Emerson (1803–1882)

“*History* ... is, indeed, little more than the register of the crimes, follies, and misfortunes of mankind.

But what experience and *history* teach is this—that peoples and governments have never learned anything from *history*, or acted on principles deduced from it.”

—Georg Wilhelm Friedrich Hegel (1770–1831)

“Racism is an ism to which everyone in the world today is exposed; for or against, we must take sides. And the *history* of the future will differ according to the decision which we make.”

—Ruth Benedict (1887–1948)