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.
Does anyone know of where I should look to find lots of good functional analysis problems? I am currently reading Kreyszig which has great commentary, but the majority of the exercises are simple.
[SOLVED] Sequences in lp spaces... (Functional Analysis)
Homework Statement
Find a sequence which converges to zero but is not in any lp space where 1<=p<infinity.
Homework Equations
N/A
The Attempt at a Solution
I strongly suspect 1/ln(n+1) is a solution.
Since ln(n+1) ->...
Hello
I need help with an analysis proof and I was hoping someone might help me with it. The question is:
Let (X,d) be a metric space and say A is a subset of X. If x is an accumulation point of A, prove that every r-neighbourhood of x actually contains an infinite number of distinct...
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Would anyone out there be able to help me with a problem I'm having? I have to prove that a function is open and that another is closed. The question is:
Consider C [0,1] with the sup metric. Let f:[0,1]→R be the function given by f(x)=x²+2
Let A={g Є C[0,1]: d(g,f) > 3}. Prove...
Hello
Would anyone out there be able to help me with a problem I'm having? I have to prove that a function is open and that another is closed. The question is:
Consider C [0,1] with the sup metric. Let f:[0,1]→R be the function given by f(x)=x²+2
Let A={g Є C[0,1]: d(g,f) > 3}. Prove...
Hi!
I was thinking about taking an introductory course in Functional analysis the commming spring, and was wondering if you more experienced guys can tell me if this is a good complement to understand theoretical quantum physics better?
Cheers
Suppose T: X -> Y and S: Y -> Z , X,Y,Z normed spaces , are bounded linear operators. Is there an example where T and S are not the zero operators but SoT (composition) is the zero operator?
I am thinking about taking a class on functional analysis. I am eventually planning on doing derivatives trading as a career. Is this class worth taking or should I try to find something more applied. I guess I am saying that I don't see how applied functional analysis is.
I have a commutative Banach algebra A with identity 1. If A contains an element e such that e^2 = e and e is neither 0 nor 1 (I think this also means to say that it contains a non-trivial idempotent), then the maximal ideal space of A is disconnected.
Currently I am trying to show this but I...
Im trying to prove the following proposition
Let (X,\|\cdot\|_X) and (Y,\|\cdot\|_Y) be normed vector spaces and let T:X \rightarrow Y be a surjective linear map.
Then T is an isomorphism if and only if there exist m,M > 0 such that
m\|x\|_X \leq \|Tx\|_Y \leq M\|x\|_X \quad \forall \, x...
Im having some difficulties proving some basic properties of the adjoint operator. I want to prove the following things:
1) There exists a unique map T^*:K\rightarrow H
2) That T^* is bounded and linear.
3) That T:H\rightarrow K is isometric if and only if T^*T = I.
4) Deduce that if T is...
Question 1
Prove that if (V, \|\cdot\|) is a normed vector space, then
\left| \|x\| - \|y\| \right| \leq \|x-y\|
for every x,y \in V. Then deduce that the norm is a continuous function from V to \mathbb{R}.
schröder's equation is a functional equation. let's assume A is a subset of the real numbers and g maps A to itself. the goal is to find a nonzero (invertible, if possible) function f and a real number r such that f\circ g=rf.
motivation: if there is an invertible f, then the nth iterate...