Functional 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. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.

Quotes

• Functional analysis emerged as an independent discipline in the first half of the 20th century, primarily as a result of contributions of S. Banach, D. Hilbert, and F. Riesz.
  • Veli-Matti Hokkanen; Gheorghe Morosanu (26 April 2002). Functional Methods in Differential Equations. CRC Press. p. 9. ISBN 978-1-4200-3536-0. 
• This tendency of functional analysis, running against arithmetization, could already be discerned in Klein's lifetime, as well as at the Hilbert school in Göttingen. Those who developed the new theories of generalized functions (Dirac in the 1920s, Sobolev in the 1930s, and Schwartz in the 1940s) were unaware of the formulations that dated back to the first decades of the 19th century. Just as in the time of Fourier, Poisson, and Cauchy, the most important applications became the partial differential equations of mathematical physics. They must be regarded as a kind of generalization of the ancient vibrating string.
  • Detlef Laugwitz (21 January 2008). Bernhard Riemann 1826-1866: Turning Points in the Conception of Mathematics. Springer Science & Business Media. p. 213. ISBN 978-0-8176-4776-6. 

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