In mathematical literature, the term 'space' is often used to describe a set endowed with additional structure, such as a metric space or a vector space. What is the historical evolution of the concept of 'space' in mathematics, specifically its formalization as a set with particular properties?" In modern books the term is not explicitly defined, even in early books of 1947, like Finite Dimensional Vector Spaces, a vector space is defined as a set with certain axioms while avoiding to define what "space" itself is.
Searching plenty of older Google Books with keywords, "the term space" or "the word space" in mathematics led to one book that explicitly opened with the following:
Linear Transformations in Hilbert Space and Their Applications to Analysis by Marshall Harvey Stone (1932)
§. The Concept of Space The word "space" has gradually acquired a mathematical significance so broad that it is rirtually equivalent to the word "class", as used in logic. Historically, the reason for" this development is to be found in the recognition that many classes which are of special importance in mathematics enjoy internal properties analogous to the familiar ones of Euclidean space. For example, the class of all continuous real functions of a real variable defined on a given closed interval can be treated as a metric space, the distance between two functions of the class being the maximum numerical value of their difference. Another class or space of peculiar importance for analysis is the class of all real functions of a real variable on a given closed interval which are Lebesgue-measurable and have Lebesgue-integrable squares; in this space, the distance between two elements or points of the space may be defined as the square root of the integral of the square of their difference. In each of these two cases, the distance between two elements or points of the space has many of the properties of distance in Euclidean geometry. Spaces such as those just described are frequently referred to as "function-spaces" or "spaces of infinitely many dimensions".
Can anyone provide a reference which shows the earliest of the term space as used in modern times? In the unabridged version of Oxford English Dictionary, the earliest use of space in mathematical sense in English is 1900 as
1900 One speaks of the geometries of metric space, of unilateral and bilateral projective space. American Journal of Mathematics vol. 22 336.
Most likely the current usage of space and structure must have come either from German or French works.
