What is Functional analysis: Definition and 115 Discussions

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. 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. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.
The usage of the word functional as a noun goes back to the calculus of variations, implying a function whose argument is a function. The term was first used in Hadamard's 1910 book on that subject. However, the general concept of a functional had previously been introduced in 1887 by the Italian mathematician and physicist Vito Volterra. The theory of nonlinear functionals was continued by students of Hadamard, in particular Fréchet and Lévy. Hadamard also founded the modern school of linear functional analysis further developed by Riesz and the group of Polish mathematicians around Stefan Banach.
In modern introductory texts to functional analysis, the subject is seen as the study of vector spaces endowed with a topology, in particular infinite-dimensional spaces. In contrast, linear algebra deals mostly with finite-dimensional spaces, and does not use topology. An important part of functional analysis is the extension of the theory of measure, integration, and probability to infinite dimensional spaces, also known as infinite dimensional analysis.

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  1. D

    Functional analysis and topology books needed

    Hi folks ... I urgently need good books about Functional analysis and Topology. These must be comprehensive and thorough, undergraduate or graduate. Please, advise and provide your experiences with such books. I accept only thick books ;) e.g Introductory Functional Analysis with...
  2. micromass

    Analysis Functional Analysis by Reed and Simon

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  3. micromass

    Analysis Essential Results of Functional Analysis by Zimmer

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    Analysis Functional Analysis by Riesz and Sz.-Nagy

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  5. micromass

    Analysis Functional Analysis by Stein and Shakarchi

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  6. micromass

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  7. micromass

    Analysis Introductory Functional Analysis with Applications by Kreyszig

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  8. T

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  9. T

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  10. K

    Convergence of Fourier Series Coefficients for L2 Functions

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  11. R

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  12. A

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  13. B

    Need help solving functional analysis problem

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  14. M

    Functional Analysis or group representations?

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  15. A

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  16. Z

    Kolmogorov & Fomin's Elements of Theory: Real Analysis or Lebesque?

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  17. A

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  18. C

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  19. W

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  20. C

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  21. I

    Functional Analysis & Others

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  22. M

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  23. B

    Functional Analysis: Open normed subspace

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  24. B

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  25. B

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  26. D

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  27. B

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  28. J

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  29. B

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  30. A

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  31. X

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  32. Fredrik

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  33. A

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  34. C

    Functional analysis applications

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  35. C

    An expression in functional analysis

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  36. I

    Functional analysis convergence question

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  37. A

    An open mapping is not necessarily a closed mapping in functional analysis

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  38. K

    Real / Functional Analysis Video Lectures?

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  39. E

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  40. T

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  41. A

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  42. W

    Functional analysis textbook recommendation needed

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  43. E

    Functional Analysis, Show that the range of a bounded linear operator

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  44. R

    Is R^w a first category topological vector space?

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  45. M

    What is Functional Analysis and How Can It Be Applied in Science?

    See the attachment.
  46. E

    Functional Analysis (Big Rudin)

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  47. Shackleford

    Elements of the Theory of Functions and Functional Analysis

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  48. R

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  49. Andy Resnick

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  50. D

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