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, as well as measure, ...

learn more… | top users | synonyms (1)

2
votes
2answers
28 views

Sequence of bounded linear operators implicating Cauchy sequence in $\mathbb K$

Let H be a Hilbert space and $(T_n)_{n \in \mathbb N}$ be a sequence in ${\rm BL}(H)$ (bounded linear operators) such that $(\langle y,T_nx \rangle)_{n \in \mathbb N}$ is a Cauchy sequence in $\mathbb ...
2
votes
1answer
31 views

Bessel sequence and synthesis operator

A sequence $(f_{k})_{k\in \mathbb{N}}$ is called a Bessel sequence in a Hilbert space $H$, if there exists $B>0$ such that $$\sum_{k\in \mathbb{N}}|\langle f,f_{k}\rangle|^{2}\leq B\|f\|^{2}$$ ...
3
votes
1answer
49 views

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 ...
2
votes
1answer
44 views

Elliptic partial differential equations and elliptic operators

I'm starting to study elliptic partial differential equations and I just want to know if there are any connections between the following concepts: An elliptic partial differential equation is given ...
0
votes
0answers
26 views

Operator Equation?

A space of polynomials $\{f_n\}$ is given, where $f_n$ is of degree $n$. The operator $T$ in this space, satisfy the relation $$T^2(f_n)+a_nT(f_n)-f_n=0$$ where $a_n$ is a scalar depending on $f_n$. ...
0
votes
1answer
27 views

boundedness in a locally convex space

From Conway, A course in functional analysis, page 107. Problem 6. Let $X$ be a locally compact space and give $C_b(X)$ the strict topology $\beta$ defined by the seminorms $p_\phi(f)= ||\phi ...
4
votes
0answers
48 views

Jordan decomposition of linear functionals

Let $X$ be a locally compact Hausdorff space. Also, let $C_0(X,\mathbb R)$ denote the vector space of such continuous functions $f:X\to\mathbb R$ that the set $\{x\in X\,|\,|f(x)|\geq\varepsilon\}$ is ...
0
votes
0answers
65 views

Functional Analysis Question

Let $(X,\|\cdot\|_1)$ and $(X,\|\cdot\|_1)$ be Banach spaces. Does it imply that $\|\cdot\|_1-\|\cdot\|_2$ (equivalent)? It is know that if $\|\cdot\|_1-\|\cdot\|_2$ and $(X,\|\cdot\|_1)$, then ...
1
vote
1answer
39 views

A Particular Frechet Derivative and Interpretation

I would like find the Fréchet derivative of the following functional: $$ \begin{align} F : C[0,1] &\rightarrow \mathbb{R}\\ w &\mapsto \frac{\int_0^1 xw(x)f(x) \, dx}{\int_0^1 w(x)f(x) \, dx}. ...
1
vote
1answer
34 views

Projecting self-adjoint operator onto closed subspace

Let $H$ be a complex Hilbert space and let $(Q, D(Q))$ be a closed, densely defined, positive semidefinite, Hermitian quadratic form on $H$. (That is, $D(Q)$ is a dense subspace of $H$, $Q$ maps ...
0
votes
1answer
26 views

Equivalence of weak-* convergence in Banach spaces.

Let $X$ be a Banach space and $f,(f_n)_{n \in \mathbb N} \in X^*$. $f_n \xrightarrow{w^*} f$ if and only if a) $\sup_{n \in \mathbb N} \|f_n\| < \infty$ and b) $\exists A \subset X: ...
0
votes
1answer
32 views

If $A$ and $B$ are diffeomorphic via $f$ and $u \in L^\infty(A)$. is $u\circ f \in L^\infty(B)$?

Let $A$ and $B$ be bounded open domains on $\mathbb{R}^n$. Let $f:B \to A$ be a diffeomorphism between $A$ and $B$. Suppose $u \in L^\infty(A)$. Is $u\circ f \in L^\infty(B)$? I think this is true, ...
2
votes
1answer
32 views

Integral over compact set and open set

Let $D \subset \mathbb{R}^d$ be open and let $a_{ij}\in L_{loc}^{1}(D\,;\mu),\,a_{ij}=a_{ji},\,1\leq i,j \leq d$ ($\mu$ is Lebesgue measure on $D$). We define $\mathcal{S}:C_{0}^{\infty}(D) \times ...
0
votes
1answer
43 views

a continuous linear functional maps the interior of a convex set to an open interval in R

I want to show that If X is a real topological vector space and $f:X\to {\Bbb R}$ is a continuous ${\Bbb R}$ - linear functional and $A$ is an open convex subset of $X$, then $f(A)$ is an open ...
0
votes
2answers
60 views

Important applications of the Uniform Boundedness Principle

There's like three applications of the uniform boundedness principle in wikipedia: 1) If a sequence of bounded operators converges pointwise to an operator, then the limit operator is also bounded, ...
2
votes
0answers
53 views

Are these functions on a Hilbert space Lipschitz equivalent?

Let $H$ be a Hilbert space with inner product $\langle\cdot,\cdot\rangle$. Fix a bounded operator $T$ on $H$, and $1\leq p<\infty$ (you can assume $p$ is an integer if necessary). Consider the ...
0
votes
1answer
31 views

Show that $L^p$ norm is logarithmically convex as a function of $p$

Let $\Omega \subset \mathbb{R}^n$ be an open, bounded domain. Let $u : \Omega \to \mathbb{R}$ be measurable with $||u||_\infty < \infty$. For $p \in [1, \infty)$, define $$\Phi_u : p \mapsto ...
2
votes
1answer
21 views

Differentiability of product/composition of function

How will be the product and composition of two functions, where one is differentiable and another is just continuous, behave?I mean to say, if the product or composition is differentiable, then what ...
1
vote
1answer
36 views

Proposed proof Lebesgue integration question

I just want to confirm the following proof: Consider a function $u: \Omega \rightarrow \mathbb{R}$ where $\Omega \subset \mathbb{R}^{n}$ and $u \in C^{2}(\bar{\Omega})$. Let $a_{jk}$ be smooth ...
3
votes
1answer
26 views

Sobolev's inequality from Reed & Simon vol. II

In Reed & Simon vol. II, an inequality called Sobolev's inequality is stated in Eq. (IX.19): Let $0<\lambda<n$ and suppose that $f\in L^p(R^n)$, $h \in L^r(R^n)$ with $p^{-1} + r^{-1} + ...
1
vote
1answer
38 views

Constructing a specific normalised test function.

Given the following class of normalised test functions: \begin{equation} \mathcal{T}=\{\phi\in C_c^{\infty}(\mathbb{R}^n)\ |\ \text{supp }\phi \subseteq B(0, 1),\ \|D\phi\|_{\infty}\leq 1\} ...
0
votes
2answers
92 views

Finding the spectrum of this operator

Let $X$ be a Hilbert space and let $\psi_1,\psi_2$ be linearly independent vectors and let $\varphi_1,\varphi_2$ be linearly independent vectors in $X$. Define the operator $T$ in $B(X)$ ...
0
votes
0answers
27 views

Follow up on star algebra (proof verification)

I previously asked this question about a proof of the following claim: If $A$ is a commutative non-unital non-zero $C^\ast$ algebra then $\Omega (A)$ is not empty. In the meantime I believe to ...
1
vote
0answers
28 views

Functional characterization of zeroth law of thermodynamics [Sepration of Variables]

Zeroth law of thermodynamics is stated also as: If A is in thermal equilibrium with B and if B is in thermal equilibrium with C, then A is in thermal equilibrium with C. This can be formulated ...
2
votes
1answer
49 views

$l_{\infty}$ is the quotient of $l_{1}(\aleph_{1})$?

I wonder if $l_{\infty}$ is the quotient of $l_{1}(\aleph_{1})$. If so, how to prove it?
2
votes
1answer
39 views

Unconditional bases equivallent to permutations of basis elements.

On page 9 of The Handbook of the Geometry of Banach Spaces: Volume I I found the following: "A basis $\{x_n\}_{n=1}^{\infty}$ is said to be an unconditional basis provided that $\sum \alpha_n x_n$ ...
0
votes
0answers
19 views

Weyl Operators: Discontinuity

Let $\mathcal{A}_{CCR}(\mathcal{H})$ be a CCR algebra over a Hilbert space $\mathcal{H}$. Then the Weyl operators are unitary and therefore $\sigma(W(f))\subseteq \mathbb{S}$ so by the spectral ...
2
votes
1answer
48 views

Implying a positive definite operator

If we are given that $A:V \rightarrow V$ is an operator where $V$ is a real Hilbert space. If we are given that $A$ is bounded, strictly positive $\big(\langle Au,u \rangle > 0$ for all $u \neq ...
-1
votes
1answer
34 views

Bogoliubov Transformation

Let $\mathcal{A}_{CAR}(\mathcal{H})$ be a CAR algebra over a Hilbert space $\mathcal{H}$. Consider a linear $S$ and an antilinear $T$ both bounded operators acting on $\mathcal{H}$ satisfying: ...
3
votes
1answer
51 views

Solution of a functional integral

I am trying to show the following integral has the following result $$-\int \nabla^2\psi \text{d} \psi^*=|\nabla\psi|^2$$ Going backwards I write ...
3
votes
3answers
46 views

Understanding the Euler operator

While reading this book I came across a differential equation $$t^5\frac{d^2y}{dt^2}+2t^4\frac{dy}{dt}-y=0$$ that was then rewritten in terms of the Euler operator, $\delta=t\frac{d}{dt}$, with the ...
0
votes
1answer
31 views

Why is $\Gamma (A)$ closed

Let $A$ be a commutative $C^\ast$ algebra and let $\Gamma: A \to C_0(\Omega (A))$ be the map $a \mapsto \widehat{a}$. Here $\Omega(A)$ denotes the character space of $A$. Why is $\Gamma (A)$ closed ...
0
votes
1answer
27 views

Another question about $C^\ast$ algebra

If $A$ is a Banach algebra then let $\Omega (A)$ denote the character space of $A$. Apparently there exist abelian Banach algebras such that $\Omega (A) = \varnothing$. Also apparently, if $A$ is a ...
0
votes
1answer
34 views

Where is commutativity of $b$ needed?

I have a question about the following proof: If $e^{ia}-e^{i\lambda}=(a-\lambda)be^{i\lambda}$ and $(a-\lambda)$ is not invertible then $(a-\lambda)x$ is not invertible for all $x$. Why "since $b$ ...
0
votes
1answer
50 views

Space of continuous functions vanishing at infinity

Let's denote with $C_0(X)$ the space of continuous functions $f$ on $X$ such that for every $\epsilon>0$ there exists a compact set $K_\epsilon\subset X$ satisfying $sup_{x\notin K_\epsilon ...
0
votes
1answer
95 views

Irrational Rotation

Let $\sigma$ be a homeomorphism of $S^1$. Then the following statements are equivalent; (1) O(z) is dense in S for some z in S, (2) O(z) is dense in $S^1$ for every z in $S^1$, (3) $\sigma$ is ...
0
votes
1answer
24 views

Dirac delta function and well behaved function [duplicate]

whether dirac delta function a well behaved function? Can u please explain the properties of a well behaved function..?
6
votes
1answer
96 views

best intuitive books/video lectures to read topology and functional analysis

What are the best intuitive books/video lectures to read topology and functional analysis ? I am aware of basic linear algebra, analysis and measure theory.
1
vote
0answers
28 views

Differentiate a log of $L^p$ norm, don't understand this result

I'm reading this paper. In it, the authors show this lemma: And then they prove this lemma My question is: I have no idea how they get the result in Lemma 3.2. Do we not get $$\frac{d}{ds}\log ...
1
vote
1answer
14 views

How to show $\mu_A(x)=\inf\{\alpha>0: \alpha^{-1} x\in A\}$ where $\mu_A(x)$ is the Minkowski functional of $A$?

Let $X$ be a $\mathbb K$-vector space ($\mathbb K=\mathbb R$ or $\mathbb C$) and suppose $A\subset X$ is convex (and absorbing). How to show $$\{x\in X: \mu_A(x)<1\}\subset A?$$ Above ...
0
votes
1answer
22 views

Check complex differentiability

I am trying to take a derivative w.r.t $z\in\mathbb{C}$ of the following map: $z\mapsto \sum_{j=0}^{\infty}\lambda_{j} (T(\psi+zh))_{j}$ where $(\lambda_{j})$ is a bounded sequence, $T$ is a ...
2
votes
2answers
32 views

Ultra weakly closed *-subalgebra of B(H)

I'm currently working on a text about von Neumann algebras and the author used without further clarifying that any ultra weakly closed *-subalgebra of $B(H)$ contains a largest projection. Could ...
0
votes
1answer
40 views

Topology of Normed Space

$(X, \lVert \cdot \rVert)$ is a normed space. Let $x \in X \setminus \{0\}$ and $Y \subset X$ is a subspace. Prove that if $Y$ is open then $Y=X$. Which technique is more useful? We know ...
1
vote
1answer
29 views

Operator Tensor Product

Let $S$ and $T$ be bounded operators over a Hilbert space $\mathcal{H}$. Define their tensor product $S\otimes T$ as acting on $\mathcal{H}\otimes\mathcal{H}$ by $S\otimes T(x\otimes y):=Sx\otimes Ty$ ...
0
votes
0answers
20 views

A locally convex space is metrizable if and only if it is first countable

I'm studying Functional Analysis by myself. the following is an exercise while I'm not sure about my answer. If $X$ is a locally convex space (LCS), show that $X$ is metrizable if and only if $X$ is ...
0
votes
1answer
26 views

Extend the Stone-Weierstrass theorem to high dimension?

I am thinking of if there is high dimensional extension to the well known Stone-Weirstrass theorem. Wikipedia says it is possible to extend the 1D theorem to 2D, i.e. If  f  is a continuous ...
1
vote
1answer
53 views

Prove that $(C_{00},\|\cdot\|)_{\ell^2}$ is not a Banach space

How can we prove that $(C_{00},\|\cdot\|)_{\ell^2}$ is not a Banach space? How can we find counter example for this problem?
1
vote
1answer
41 views

Finite-dimensional subspace normed vector space is closed

I know that probably this question has already been answered, but I'd like to present my attempt of solution. Let $(E,\|\cdot\|)$ be a normed vector space and let $F\subseteq E$ be a ...
3
votes
1answer
29 views

Every unitary representation of a compact group is a direct sum of irreducible representations.

I've read nice proofs of a few different variants of the Peter-Weyl theorem and its corollaries. For instance I know that for $G$ a compact group, $L^2(G)$ is a Hilbert space direct sum of the matrix ...
1
vote
1answer
71 views

Any idea with this problem of distances???

Let $E$ a normed linear space and $H$ the closed hyperplane $H=\ker f$, where $f\in L(E,\mathbb{R})$, $f\not\equiv 0$. Show that if $a\in E$ then $$d(a,H)=\frac{|f(a)|}{||f||}$$ And the problem have a ...