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Questions tagged [functional-analysis]

Functional analysis, the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces and other topics. For basic questions about functions use more suitable tags like (functions), (functional-equations) or (elementary-set-theory).

Functional analysis is the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces, including spectral theory, as well as measure, integration, probability on infinite dimensions, and also manifolds with local structure modeled by these vector spaces.

For basic questions about functions use more suitable tags like , or .

52582 questions
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Shorter proof for $T$ compact and $x_n \to x$ weaky then $Tx_n \to Tx$ strongly

I proved that if $X,Y$ are Banach spaces and $T: X \to Y$ is compact and $x_n \to x$ weakly then $Tx_n \to Tx$ strongly. I am now wondering if there is a shorter proof? Here is my proof: Let $x_n \to x$ weakly. By this result here $T$ is norm-norm…
user167889
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E is bounded if and only if every countable subset of it is bounded

I'm trying to do an exercise from Rudin's "Functional analysis, 2nd edition". It is question 6 from the first chapter: "Prove that a set E in a topological vector space is bounded if and only if every countable subset of E is bounded" My efforts so…
MathsStudent
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Basic Question about notation in the space of continuous functions

I am reading the book "Introduction to Calculus of Variations" by Bernard Dacorogna (Could not find a link in google books) where he defines $C(\bar{\Omega})$ to be the space of continuous functions $u : \Omega \to \mathbb{R}$ which can be…
jpv
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Spectrum of $\int\limits_0^x f(t) dt$ operator

Let $A\colon E\to E$ definied by $A(f)(x)= \int\limits_0^x f(t) dt$. I have to find the spectrum of $A$ in the cases $E=C[0,1]$ and $E=L_2[0,1]$. I have proved that $A$ has no eigenvalues, but I can't find full spectrum.
Philipp G. Sinicyn
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Prove that $Af(x) = \frac 1x \int\limits_0^x f(t) dt$ isn't compact in $L_2[0,1]$

I need to prove that operator $\displaystyle Af(x) = \frac 1x\int\limits_0^x\ f(t) dt$ isn't compact in $L_2[0,1]$. I have tried to calculate spectrum of $A$, but failed.
Philipp G. Sinicyn
  • 891
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Show that $\lambda \in \sigma(A),$ $\lambda$ not an eigenvalue, implies that $\lambda \in \sigma(A + K)$ where $K$ is compact.

Let $A : H \rightarrow H$ be a bounded linear map where $H$ is a Hilbert space with $\dim H = \infty$. Suppose that $\lambda \in \sigma(A)$ but $\lambda$ is not an eigenvalue. Let $K : H \rightarrow H$ be compact. Show that $\lambda \in \sigma(A +…
Daniel Watkins
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Showing the set $A+B$ is closed.

Let $X$ be a banach space, and let $A$, and $B$ be closed linear subspaces. Assume that $$\inf\{\|x-y\|\mid x\in A, y\in B, \|x\|=\|y\|=1\}>0$$ I want to show that $A+B$ is closed. I was thinking of doing something like, let $z$ be a limit point of…
Daniel Montealegre
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What is common between adjoint operator and transpose of the matrix?

I am confused. What is Connection between Adjoint Operator and transpose of the Matrix? I will be very grateful if someone can help me to clarify it.
learner
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Direct sum of compact operators is compact

I have that $T_n$ are bounded operators on $H_n$ ($n\geq 1$) and that $\sup ||T_i||<\infty$. Define $T=\oplus T_n$ and $H=\oplus H_n$. I want to show that $T$ is compact iff $T_n$ is compact for all $n$ and $||T_n||\rightarrow 0$. Here is what I…
Daniel Montealegre
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Is the space of compactly supported $k$-times differentiable functions separable?

Is the space $C^k_c(\mathbb{R}^n)$ of compactly supported $k$-times continuously differentiable functions with the norm $\|f\|:=\sum_{|\alpha|\leq k}\|D^\alpha f\|_\infty$ separable?
user100322
  • 93
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Show that if operator $T$ is such that $||I-T||<1$ , then $T$ is bijective.

I came across this statement in a proof and I can't figure out why its true, could someone point out why (or give a hint). Thanks. Suppose that $T:X\to X$ is a bounded linear operator that maps a Banach space $X$ to itself such that…
jkn
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$ (\mathbb{R}^3, \|.\|_1) $ and $ (\mathbb{R}^3, \|.\|_\infty) $ cannot be isometric

Can anyone prove that the spaces $ (\mathbb{R}^3, \|.\|_1) $ and $ (\mathbb{R}^3, \|.\|_\infty) $ cannot be isometric? Thanks.
Axiom
  • 433
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subset relations among Sobolev spaces and their duals

This may be a rather dense question, but I would nevertheless be grateful for some guidance. The question has to do with the Sobolev spaces $H^m (\Omega)$ on an open bounded domain $\Omega$ of $\mathbb{R}^n$, where $m \geq 0$ is integer. These…
Mark Rashid
  • 53
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to prove infinity norm is indeed norm

to prove infinity norm of a function which is equal to supremum of absolute value of that function, is indeed a norm. I have a clue to check for the three axioms of norm. but tell me hoe to proceed
sara
  • 93
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Help proving that a Banach space is reflexive

I'm having trouble in proving that the following space is reflexive: $$E = \{ x= (x_n) : x_n \in \mathbb{R}^n \text{ and } \sum \|x_n\|^2_\infty < \infty\}$$ with the norm $$ \|x\| = (\sum \|x_n\|^2_\infty)^\frac{1}{2}$$ I already tried an analogous…
Willian Hans
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