In contrast, linear algebra deals mostly with finite-dimensional spaces, and does not use topology. In modern introductory texts on functional analysis, the subject is seen as the study of vector spaces endowed with a topology, in particular infinite-dimensional spaces. Hadamard also founded the modern school of linear functional analysis further developed by Riesz and the group of Polish mathematicians around Stefan Banach. The theory of nonlinear functionals was continued by students of Hadamard, in particular Fréchet and Lévy. However, the general concept of a functional had previously been introduced in 1887 by the Italian mathematician and physicist Vito Volterra. The term was first used in Hadamard's 1910 book on that subject. The usage of the word functional as a noun goes back to the calculus of variations, implying a function whose argument is a function. This point of view turned out to be particularly useful for the study of differential and integral equations. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis.įunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g.
One of the possible modes of vibration of an idealized circular drum head.
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