Geometry, a world of ideal forms created by the Greek mathematicians, underwent an analytical transformation with the introduction of coordinates in the 17th century, notably by René Descartes. In 1804, Bernard Bolzano introduced ideas that foreshadowed vector calculus, which Giusto Bellavitis (1803–1880) later established through his "calculus of equipollences." Giuseppe Peano's modern definition of vector spaces was developed in the early 20th century by mathematicians such as Poland's Stefan Banach and Hungary's Frigyes Riesz. They founded functional analysis, in which functions play the role of vectors.
A geometry of functions
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Around that time, John von Neumann introduced Hilbert spaces, function spaces that may be infinite-dimensional. This fundamental structure applies equally well to vectors in geometry of any dimension and to spaces of sequences, polynomials or solutions to differential equations. These spaces inherit the features and vocabulary of vector geometry and are therefore equipped with an inner product (see box). Here we consider only Euclidean spaces, namely vector spaces over the field of real numbers.
Functions, the vectors in these function spaces, must be closed under linear combinations. Continuous functions, for example, have this property: if f and g are two such functions and α and β are real numbers, then αf + βg is itself continuous. Integration, a linear operation that assigns a scalar to a function, is a natural candidate for defining an inner product. For example, the following symmetric bilinear form gives an inner product on the space of real-valued continuous integrable functions on an interval I of the real line:
