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. 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. 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. 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. 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. 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?

8. 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. 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. 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. 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. 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: $$||f(x)||\geq g\left( \text{dist}(x,\mathcal{X})\right),$$ with ##\mathcal{X}:= \{x:f(x)=0\}## (zero...
13. 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. 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. 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 ?

24. 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. 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. 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 ?

49. 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. 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...