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. P

    A Functional Analysis for Physics in 2024

    Physicists provided the motivation for studying functional analysis (FA) 100 years ago. But is an in depth understanding of FA necessary in 2024? A slightly different way of putting it would be: is there any important work being done by physicists that requires working knowledge of all the...
  2. P

    Show that a set has no "unique nearest point" property

    From Bridges' Foundations of Real and Abstract Analysis. I'm given the following hint. Given ##a=(a_n)\in c_0\setminus S##, set ##\alpha=\sum _{n=1}^{\infty }2^{-n}a_n## and show that ##d(a,S)\leq|\alpha|##. Let ##x=(x_n)\in S##, suppose that ##\lVert a-x\rVert\leq|\alpha|##, and obtain a...
  3. DAntanov

    Writing L-functions in Python (or any other language)

    Not many code examples exist for how one would go about writing an L-function. Can anyone give me a step-by-step run down of how to do this and/or link me to relevant resources?
  4. MexChemE

    Analysis Study plan for Functional Analysis - Recommendations and critique

    Hello, PF! It’s been a while since I last posted. I am looking for a critique and recommendations regarding my study plan towards Functional Analysis and applications (convex optimization, optimal control), but first, some background: - This plan is in preparation for my master’s thesis, I...
  5. G

    Functional Analysis exchange year at Imperial

    Hey, I would like to do an exchange year at Imperial. I would like to follow as a physicist the Functional Analysis course. However, I have not heard the best things about this peculiar course. What is the audience opinion on that?
  6. H

    Matrix with a bounded mapping as an entry is bounded

    In a previous exercise I have shown that for a $$C^{*} algebra \ \mathcal{A}$$ which may or may not have a unit the map $$L_{x} : \mathcal{A} \rightarrow \mathcal{A}, \ L_{x}(y)=xy$$ is bounded. I.e. $$||L_{x}||_{\infty} \leq ||x||_{1}$$, $$x=(a, \lambda) \in \mathcal{\hat{A}} = \mathcal{A}...
  7. H

    Compact Hausdorff Spaces: Star-Isomorphic Unital C*-Algebras

    If ##X## and ##Y## are homeomorphic compact Hausdorff spaces, then ##C(X)## and ##C(Y)## are ##star##-isomorphic unital ##C^{*}##-algebras. So I got the following map to work with (AND RECALL THAT ##C(X)## and ##C(Y)## are vector spaces). $$C(h) : C(Y) \rightarrow C(X) \ : \ f \mapsto f \circ...
  8. H

    Bounded operators on Hilbert spaces

    I have to show that for two bounded operators on Hilbert spaces ##H,K##, i.e. ##T \in B(H)## and ##S \in B(K)## that the formula ##(T \bigoplus S) (\alpha, \gamma) = (T \alpha, S \gamma)##, defined by the linear map ##T \bigoplus S: H \bigoplus K \rightarrow H \bigoplus K ## is bounded...
  9. D

    A Applications of analysis in signal processing/machine learning?

    Hello everyone, My question for this thread concerns the application of (mainly) mathematical analysis to fields such as signal processing and machine learning. More specifically, I was wondering if you happen to know of some interesting application of things like measure theory or functional...
  10. Yellotherephysics

    A Functional Determinant of a system of differential operators?

    So in particular, how could the determinant of some general "operator" like $$ \begin{pmatrix} f(x) & \frac{d}{dx} \\ \frac{d}{dx} & g(x) \end{pmatrix} $$ with appropriate boundary conditions (especially fixed BC), be computed? And assuming that it diverges, would it be valid in a stationary...
  11. V

    A Closure of constant function 1 on the complex set

    I'm watching this video to which discusses how to find the domain of the self-adjoint operator for momentum on a closed interval. At moment 46:46 minutes above we consider the constant function 1 $$f:[0,2\pi] \to \mathbb{C}$$ $$f(x)=1$$ The question is that: How can we show that the...
  12. V

    A What type of function satisfy a type of growth condition?

    Let ##f:\mathbb{R}^n\rightarrow\mathbb{R}^n##. Is there any class of function and some type of "growth conditions" such that bounds like below can be established: \begin{equation} ||f(x)||\geq g\left( \text{dist}(x,\mathcal{X})\right), \end{equation} with ##\mathcal{X}:= \{x:f(x)=0\}## (zero...
  13. L

    A Definitions of Cylinder Sets and Cylinder Set Measure

    I'm trying to learn about Abstract Wiener Spaces and Gaussian Measures in a general context. For that I'm reading the paper Abstract Wiener Spaces by Leonard Gross, which seems to be where these things were first presented. Now, I'm having a hard time to grasp the idea/motivation behind the...
  14. M

    A What do the notations in functional analysis mean for a given function?

    Hi PF! Can someone help me understand the notation here (I've looked everywhere but can't find it): given a function ##f:G\to \mathbb R## I'd like to know what ##C(G),C(\bar G),L_2(G),W_2^1(G),\dot W_1^2(G)##. I think ##C(G)## implies ##f## is continuous on ##G## and that ##C(\bar G)## implies...
  15. A

    Mathematics behind Signal and Systems

    I am looking for a signal processing textbook that uses real, complex, and functional analysis with measure theory. In other words, mathematically rigorous signal processing. Specifically, I prefer the kind that takes time to review all the topics from mathematical analysis before jumping into...
  16. S

    I Norm of a Functional and wavefunction analysis

    Hi, I am working on a home-task to analyse the properties of a ODE and its solution in a Hilbert space, and in this context I have: 1. Generated a matrix form of the ODE, and analysed its phase-portrait, eigenvalues and eigenvectors, the limits of the solution and the condition number of the...
  17. Jianphys17

    I The role of the weight function for adjoint DO

    Hi at all, I've a curiosity about the role that the weight function w(t) she has, into the define of adjoint & s-adjoint op. It is relevant in physical applications or not ?
  18. R

    A Solution of a weakly formulated pde involving p-Laplacian

    Let $$f:\Omega\to\mathbb{R}$$, where $$\Omega\subset\mathbb{R}^d$$, and $$\Omega$$ is convex and bounded. Let $$\{x_i\}_{i=1,2,..N}$$ be a set of points in the interior of $$\Omega$$. $$d_i\in\mathbb{R}$,$i = 1,2,..N$$ I want to solve this weakly formulated pde: $$ 0=\frac{A}{N^{d+1}} \sum_i...
  19. Jianphys17

    Rigor in Quantum physics -- Do I need to know Functional Analysis well?

    Hello, I've a following question: Is necessary know well func. analysis, and all its theorems to handle well quantum physics...?
  20. KennethK

    Calculate the spectrum of a linear operator

    <mod note: moved to homework> Calculate the spectrum of the linear operator ##T## on ##B(\ell^1)##. $$T(x_1,x_2,x_3,\dots)=(\sum_{n=2}^\infty x_n, x_1, x_2, x_3, \dots)$$I think the way to do it is to find the point spectrums of ##T## and its adjoint ##T^*##. But I don't know how to calculate...
  21. J

    Argue, why given Operators are compact or not.

    Which of the operators T:C[0,1]\rightarrow C[0,1] are compact? $$(i)\qquad Tx(t)=\sum^\infty_{k=1}x\left(\frac{1}{k}\right)\frac{t^k}{k!}$$ and $$(ii)\qquad Tx(t)=\sum^\infty_{k=0}\frac{x(t^k)}{k!}$$ ideas for compactness of the operator: - the image of the closed unit ball is relatively...
  22. Jianphys17

    Courses Gd of curve and surfaces or functional analysis before?

    Hello everyone, i just finished a course of analysis(2)\vector calculus.Now iI'm interested in doing Gd of curves and surfaces(Do Carmo), and functional analysis(Rudin'sbook), but do not know what may have precedence between the two, on which i should start before you think?
  23. H

    I How to find the inverse of an integral transform?

    I'm trying to find the distribution of a random variable ##T## supported on ##[t_1, t_2]## subject to ## \mathbb{E}[V(t', T)] = K, \forall t' \in [t_1, t_2]##. In integral form, this is : $$ \int_{t_1}^{t_2} V(t', t).f(t) \, dt = K,\forall t' \in [t_1, t_2], $$ which is just an exotic integral...
  24. J

    Looking for Mathematically Rigorous QM Textbooks? Any Suggestions?

    Upon searching in this forum, i have found discussions about the standard undergraduate textbooks on QM not being so good in teaching you the foundations properly. A good example is the difference between Hermitian and self-adjoint operators. Some people are saying that we should study QM from a...
  25. S

    MHB Help with functional analysis questions

    Hi, Could someone post a solution to the following questions : 1. Let R be the real numbers and A a collection of all groups that are either bound or their complement is bound. a. Show that A is an Algebra. Is it a sigma algebra? b. Define measure m by m(B) = {0 , max(on B) x <...
  26. evinda

    MHB Applied Functional Analysis: What You Need to Know

    Hello! (Wave) What is Applied Functional Analysis about? What knowledge is required? Which is the difference between this and Functional Analysis ?
  27. S

    MHB Help for test - functional analysis

    Hi - my professor in functional analysis posted 4 prior years tests just 4 days before the test without solutions. I'd appreciate if anyone can help send solutions for the following with the following questions : 1. $\mu$ is a sigma additive measure over sigma algebra $\Sigma$. A $\in...
  28. S

    MHB What is a Semigroup and How Does it Relate to Immeasurable Sets?

    Hi, I'm taking a course in functional analysis and having some trouble with the following questions : 1. L1(R) is the space of absolutely integrable functions on R with the norm integrate(abs(f(x)) over -inf to +inf. Define a linear operator from L1(R) to L1(R) as A(f)(x)=integrate...
  29. B

    Analysis Which book will suit the following course syllabus (introductory analysis)?

    Dear Physics Forum personnel, I am a undergraduate student with math and CS major who is currently taking an introductory analysis course called MATH 521 (Rudin-PMA). On the next semester, I will be taking the course called MATH 522, which is a sequel to 521. My impression is that 522 will be...
  30. F

    Functional analysis, ortho basis, weakly convergent

    Homework Statement This is a problem from Haim Brezis's functional analysis book. Homework EquationsThe Attempt at a Solution I'm assuming (e)n is the vectors like (e)1 = (1,0,0), (e)2=(0,1,0) and so on. We know every hilbert space has an orthonormal basis. I also need to know the...
  31. S

    Deviation after completion of the doctorate

    At the moment I'm in the final stages of my doctorate in mathematics. (My background is a BS in physics and an MS in mathematics.) My focus and interest have been in applied functional analysis in general and various kinds of abstract and concrete delay equations in particular. These are...
  32. AXidenT

    Functional Analysis or Differential Geometry?

    I'm in my last semester of my undergraduate majoring in mathematics (focusing on mathematical physics I guess - I'm one subject short of having a physics major) and am wondering, largely from a physics perspective if it would be better to do a functional analysis course or a differential...
  33. ddd123

    Sterman's QFT - 2.7b (on functional derivatives)

    I've been trying to fill in my mathematical blanks of things I just took as dogma before. Especially, not having a background in functional analysis, the functional derivatives often seem to me mumbo jumbo whenever things go beyond the "definition for physicists". In particular I tried looking...
  34. R

    What the terms orthogonal & basis function denote in case of signals

    I am a beginer. I have read that any given signal whether it simple or complex one,can be represented as summation of orthogonal basis functions.Here, what the terms orthogonal and basis functions denote in case of signals? Can anyone explain concept with an example?Also,what are the physical...
  35. S

    Should I Study Functional Analysis or Calculus on Manifolds?

    I have the opportunity to pursue an independent study in functional analysis (using Kreyszig's book) or calculus on manifolds (using Tu's book) next semester. I think that both of the subjects are interesting and I would like to study them both at some point in my life, but I can only choose one...
  36. L

    Classical I need a fluid mechanics textbook

    Hello, everyone, please forgive me for my poor English. I'm a sophomore, major in Astronomy. I've finished Hassani's book(Mathematical Physics). And I've learned Real variable function and functional analysis (I do not know what exact name of this course) I'd like to buy a textbook with...
  37. J

    Why there's no L^2[-inf,inf] space?

    Hello, Maybe it's a silly question, but why the space ##L^2[a,b]## has always to have bounded limits? Why can't we define the space of functions ##f(x)## where ##x \in \mathbb{R}## and ##\int_{-\infty}^\infty |f(x)|^2 dx \le M## for some ## M \in \mathbb{R^+}##? As far as I know the sum of two...
  38. J

    Which is the space of mappings from L^2 to itself?

    Hi there, I was wondering, which is the space of (not necessarily linear) mappings from ##L^2## to itself? If you have an element ##f(x) \in L^2##, then a nonlinear mapping could be ##g(\cdot)##. Then if ##g## is bounded the image is in ##L^2##, does that mean that the space of linear and...
  39. F

    Functional Analysis vs. Complex Analysis?

    I have one slot to fill in in the coming term. The two candidates are Functional Analysis and Complex Analysis (both on the undergraduate level). Here are some questions: 1) Which one would you pick and why? 2) What other classes in the standard B.Sc. math curriculum rely on either of these...
  40. D

    Functional Analysis book for beginners

    Hi, I need a functional analysis book. I have Kreyszig's book. I'm at continuous mapping but I have some problems with completeness and accumulation points. So I would like to read a lot excercises about these introductory stuff. What are your suggestions? Thanks.
  41. T

    Need a Functional Analysis book

    I need a measure/integration theory book that covers the basics. I had already calculus, complex analysis, ODEs and topics of PDEs/Sturm-Liouville problem. More specifically I need to learn functional analysis to be prepared for stochastic calculus. Any suggestions? Thank you.
  42. M

    Functional analysis: Shoe set is not dense in C([a,b])

    Homework Statement Let [a,b] \subset \mathbb{R} be a compact interval and t0 \in [a,b] fixed. Show that the set S = {f \in C[a,b] | f(t_0) = 0} is not dense in the space C[a,b] (with the sup-norm). Homework Equations Dense set: http://en.wikipedia.org/wiki/Dense_set sup -...
  43. R

    Functional analysis Gateaux & Frechet derivatives)

    Homework Statement https://imagizer.imageshack.us/v2/622x210q90/833/sqaw.png I am having difficulty understanding the notation <h, f''(x0)h>
  44. boneh3ad

    Recommended texts for self-study [real & functional analysis]

    Hello, I have been increasingly running into topics in my field where at least a basic faculty with real and functional analysis would be quite helpful and I would like to go about self-studying a bit in that area. I know that Rudin is the canonical text in the field, but I have also heard...
  45. G

    Functional analysis and real analysis

    In my schools functional analysis course, under prerequisites, it says "real analysis would be a good preparatory course, but is not required". In the concurrent real analysis thread, it was mentioned that real analysis is a stepping stone to functional analysis. I'm curious about two things...
  46. K

    How can I prove that this 2D expression is always less than 1 for n≠β?

    Hello, While analysing the asymptotic value of a ratio of a bessel and a hankel function, I reduced it to something of the form [(1 + β/n)^ n * (1 + n/β)^ β] / 2^(n+β) ; n and β are integers and greater than 1 how do I show that the above expression is always less than 1, for n≠β...
  47. W

    How to learn functional analysis

    I major in physics, but I'm also very interested in mathematics, especially analysis. Until now, I have taken mathematical analysis and real analysis. Now, I want to learn functional analysis by myself, and my teacher adviced me to read topology first. But I found it difficult to understand and...
  48. M

    MHB Troubling contradiction in Functional Analysis

    Hello I was doing an exercise that said: "If $P$ is a continuous operator in a Hilbert space $H$ and $P^2=P$ then the following five statements are equivalent". The first statement was that P is an orthogonal projection. Now this was suposed to be equivalent, under the condition of $P^2=P$, to...
  49. F

    Role of real & functional analysis in physics?

    I know complex analysis is of immense help in physics at it aids us in calculating certain integrals much more easily. But what about real analysis and functional analysis? Are these branches of mathematical analysis of much use in physics? If so, in what branches of physics and how?
  50. Hercuflea

    Numerical analysis vs functional analysis vs statistics for engineerin

    Hey all, back with another question. I have the opportunity in the fall to choose 1 (maybe 2 if I'm lucky) of the following classes: Numerical analysis (undergrad numerical linear algebra, using matlab), Functional Analysis (as a directed study course with a prof), and the other is doing a...