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, ...

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259 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 ...
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1answer
46 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
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1answer
77 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} + ...
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1answer
54 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\} ...
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2answers
136 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 $B(X)$ ...
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0answers
35 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 ...
3
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1answer
62 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
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1answer
64 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 $\sum \alpha_n x_n$ ...
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1answer
36 views

Weyl Operators: Spectrum

Given a CCR-algebra $\mathcal{A}_{CCR}(\mathcal{H})$ over a Hilbert space $\mathcal{H}$. Then the Weyl operators are unitary: $$W(f)^*=W(-f)=W(f)^{-1}$$ Thus, their spectrum lies on the unit circle: ...
2
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1answer
115 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 ...
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1answer
112 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
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1answer
57 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
105 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 ...
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1answer
37 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 ...
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1answer
40 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
381 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 ...
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1answer
164 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 ...
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1answer
38 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..?
9
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1answer
712 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.
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0answers
39 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
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1answer
42 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 ...
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1answer
43 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
57 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
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1answer
46 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 ...
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1answer
98 views

Tensor Product: Preliminary

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 $S\otimes T(x\otimes y):=Sx\otimes Ty$ ...
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0answers
134 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 ...
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votes
1answer
198 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
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1answer
82 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?
4
votes
1answer
2k 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
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1answer
66 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 ...
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1answer
85 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 ...
2
votes
1answer
76 views

How can I prove that $f$ is continuous at $0$?

Let $E$ and $F$ linear normed spaces, and consider a linear function $f:E\to F$ which satisfies for all $(x_n)\in E^{\mathbb{N}}$ such that $x_n\to 0$ then $f(x_n)$ is bounded in $F$. Then I have to ...
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votes
1answer
22 views

Showing $\int_{-1}^1\int_{-1}^1(u_x^2+2u_y^2+u^2-x^2y^2u)\, dx\, dy\geq c$.

Prove that for some $c\in\mathbb{R}$: $$G(u) =\int_{-1}^1\int_{-1}^1(u_x^2+2u_y^2+u^2-x^2y^2u)\, dx\, dy\geq c$$ for every $u \in H_0^1$. I know that $$G(u) ...
0
votes
1answer
52 views

Dual cone is a cone

Let $E$ be a Banach space and $P$ be a cone. A nonempty convex closed set $P\subset E$ is called a cone if it satisfies the conditions: $x\in P,$ $\lambda \geq 0$ implies $\lambda x\in P$; $x\in P,$ ...
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1answer
72 views

Pseudospectrum of an $n\times n$ matrix has at most n connected components

For $\epsilon>0$, the $\epsilon$ pseudospectrum of an $n\times n$ matrix $A$ is given by $\sigma_{\epsilon}(A)=\{z\in \mathbb{C}:\|(z-A)^{-1}\|>\epsilon^{-1}\}$, with the convention that ...
1
vote
1answer
97 views

Polar decomposition in a von Neumann algebra

Let $M \subseteq B(H)$ be a von Neumann algebra and $T \in M$. If $T=U|T|$ is the polar decomposition of T, why is $U \in M$? I'm thinking it's because $M$ is SOT-closed, but I'm not entirely sure.
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1answer
46 views

Projections in Tensor Product

Let $X$ and $Y$ be Banach spaces (algebras). If $X\otimes^\pi Y$ denotes the projective tensor product of $X$and $Y$, define $P:X\otimes^\pi Y\rightarrow X$ as follows : for $x\otimes y\in X\otimes ...
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votes
2answers
113 views

a non-decreasing sequence of functions with bounded L^p-norm is a Cauchy sequence in L^p space

Let $\{f_k\}_{k=1}^\infty$ be a sequence in $L^p(\mathbb{R})$ for $1\leq p<\infty$. Suppose $f_1\leq f_2\leq\cdots$ and $\sup ||f_k||_p<\infty$. Prove that $\{f_k\}_{k=1}^\infty$ converges in ...
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2answers
252 views

Analytic Vectors (Nelson's Theorem)

Is there a (simple) proof for Nelson's theorem that a symmetric operator is essentially selfadjoint if it contains a dense subset of analytic vectors?
4
votes
1answer
169 views

Estimating the sum of a series ($\ell^1$ norm) in terms of two weighted $\ell^2$ norms

Does there exist a $C > 0$ such that $$ \sum_{n \geq 1} a_n \leq C \left( \sum_{n \geq 1} 2^n a_n^2 \right)^{1/4} \left( \sum_{n \geq 1} 2^{-n} a_n^2 \right)^{1/4} $$ for all $a_n \geq 0$ with ...
0
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1answer
74 views

why is this a contraction map?

Let $B$ be a Banach space and $V$ be a normed linear space and $L_0$, $L_1$ be bounded lienar operators from $B$ to $V$. For each $t\in[0,1]$, set $$L_t=(1-t)L_0+tL_1$$ and suppose that there is a ...
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votes
2answers
81 views

Weak solution Boundary Value problem

I have to prove that the following problem $$(P) \begin{cases} -u''-u=1\,\,\,\,\,\,\,\,\text{if}\,\,\, x\in(0,\pi)\\ u(0)=u(\pi)=0 \end{cases} $$ doesn't admit weak solutions. I'm proceeding by ...
5
votes
1answer
182 views

If $h$ is closer to $f$ than to $g$, its integral on $\{f > g\}$ must “agree” with $f$'s?

I have the following question (a result I would like to prove, with an admirable record of failing at it so far) -- I actually do not know if it is obvious or just plain wrong. Let $f,g,h\colon ...
2
votes
1answer
56 views

Vice-versa of the Riesz representation theorem

I was wondering about the vice-versa of the Riesz representation theorem. In the form that was presented to me, the theorem states that if $\phi(x): H\rightarrow\mathbb{C}$ is a continuous linear ...
0
votes
2answers
106 views

How to prove this map is injective

Let $A$ be a non-unital $C^\ast$ algebra and let $M(A)$ denote the multiplier algebra and let $\widetilde{A}$ denote the unitisation of $A$. Consider the map $\varphi : \widetilde{A}\to M(A)$ ...
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2answers
124 views

Positive Elements: Characterization

Problem Given a C*-algebra with unit $1\in\mathcal{A}$. Define positive elements as: $$A\geq0\iff\sigma(A)\geq0\quad(A=A^*)$$ Positive elements can be characterized by: $$A\geq0\iff A=B^*B$$ ...
2
votes
2answers
47 views

every compact subset of a TVS is bounded

I'm self-studying Functional Analysis. The following is Exercise 4.2.4 of Conway's Functional Analysis. Let $X$ be a topological vector space. Show that every compact subset of $X$ is bounded. For ...
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vote
1answer
121 views

Approximating continuous function by the span of $\{\sin(nx)\}$

Let $f $ be continuous function on $[0,2\pi ]$ and $\int_0^{2\pi} f(x) \sin(nx) \, dx =0$ for all $n$ then prove that $f$ is identically zero. Some of my friends claim that it is not true just ...
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1answer
89 views

Application Closed Graph Theorem to Cauchy problem

Consider $E:=C^0([a,b])\times\mathbb{R}^n$ and $F:=C^n([a,b])$ equipped with the product norms. Consider $$ u^{(n)}+\sum_{i=0}^{n-1}a_i(t)u^{(i)}=f $$ with $$u(t_0)=w_1,\dots,u^{(n-1)}(t_0)=w_n \\ ...