In the modern development of functional analysis, inspired by the pioneering work of David Hilbert, Erhard Schmidt and Friedrich Riesz, the appearance of J. von Neumann (1928) signaled a decisive change. Before him the theory of linear operators and of quadratic and Hermitian forms was tied in an essential way to the coordinate representation of the vector spaces considered and to the matrix calculus. Von Neumann’s investigations brought about an essentially new situation. Linear and quadratic analysis were freed from these restrictions and shaped into an “absolute” theory, independent of coordinate representations and also largely of the dimension of the vector spaces. It was only on the basis of the general axiomatic foundation created by von Neumann that the geometric points of view, crucial to Hilbert’s conception of functional analysis, were able to prevail. It is not necessary here to recall in more detail the enormous development to which von Neumann’s ideas opened the way.
KeywordsDifferential Calculus Coordinate Representation Finite Dimensional Space Quadratic Analysis Decisive Change
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