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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1answer
103 views

Can I easily deduce this stronger spectral theorem from this weaker one?

I've just read a nice proof that: For $T$ a self-adjoint bounded linear operator on a Hilbert space $E$, there exists a unique $C^*$-algebra isomorphism $C(\sigma (T)) \rightarrow A_T$, from the ...
-1
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1answer
30 views

Number Operator closable on Fock Space?

In Bratelli Robinson the number operator in Fock space is defined as: $$\mathcal{D}(N):=\{\phi\in\mathcal{F}:\sum_{n=1}^\infty n|\|\phi_n\|<\infty\}\\ N:\mathcal{D}(N)\to ...
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0answers
21 views

Spectral Representation for a real valued process

So I just finished reading a section in a book which discusses how every stationary stochastic process $\xi(t)$ can be expressed as $\xi(t)=\int_{\mathbb{R}}e^{it\lambda}\,dZ(\lambda)$ where ...
2
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0answers
49 views

Show that $e^{-a|x|}$ does not belong to Schwartz space

Let $f : \mathbb R \to \mathbb R$ and $a > 0$ given by $f(x) = e^{-a|x|}$. Show that $f$ is rapidly decreasing and belongs to $L_1(\mathbb R)$, but not to $\mathcal S(\mathbb R)$. I had shown that ...
4
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1answer
43 views

Surjection of norms

Let $V$ be an infinite dimensional $\mathbb{C}$ (or $\mathbb{R}$) vector space. Suppose there exists two norms on $V$ such that \begin{equation*} \| \cdot\|_1 \leq \| \cdot \|_2. \end{equation*} Is ...
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0answers
12 views

Different definitions of Morrey and Campanato Spaces

The book by Giaquinta defines Campanato spaces using the seminorm: $$[u]_{p,\lambda} = \left(\sup_{\substack{{x_0\in\Omega \\ ...
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1answer
39 views

In unital Banach algebra $r(a^n) = (r(a))^n$

I tried to prove the following: If $A$ is a unital Banach algebra and $r(a)$ denotes the spectral radius then $r(a^n) = (r(a))^n$. Could somebody please tell me if I got this proof right? Thanks. ...
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1answer
64 views

Weak Derivative Heaviside function

I have to prove that the Heaviside function $$ H(x):=\begin{cases} 1 &\mbox{if } x \in [0,+\infty) \\ 0 &\mbox{otherwise}\end{cases} $$ doesn't admit weak derivative in ...
6
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1answer
144 views

Looking for different proofs of “Discrete Liouville's Theorem”.

Good day. There is a question I have already encountered twice, in very different contexts, that is relatively simple looking, but both solutions I know involve some pretty advanced theorems from the ...
0
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0answers
17 views

meagre and residual set

Assume we have a complede metric space. Then there is no meagre and residual subset. I don't really see the conclusion of the proof. Let's call $A$ this subset and assume that it is residual and ...
1
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2answers
63 views

Can all functions be expressed in terms of elementary functions?

After being introduced to the non-elementary function through an attempt to evaluate $\int x \tan (x)$, an interesting question occurred to me: Can the non-elementary functions be decomposed to ...
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1answer
33 views

Approximating function that vanishes at $0$ and $1$

In $C[0,1]$, let $\mathcal A=$ Span$\{ x^n(1-x):n\ge 1\}$, Prove that $\mathcal A$ is an algebra whose uniform closure is $\{ f\in C[0,1]:f(0)=f(1)=0 \} $. I know how to show an algebra, for the ...
0
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1answer
20 views

Example of completely positive map from Mn to Mm

I can't find any example could someone please show me one because I am trying to understand completely positive maps
1
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1answer
52 views

Approximation of the Heaviside Function whose derivative has a compact support

I am looking for a smooth approximation $H_\delta$ of the Heaviside function, which has the property that $$ \lim_{\delta\rightarrow 0^+}H_\delta =H $$ in the distribution sense, and $$ ...
1
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1answer
24 views

If $u \in L^2(0,T;H^1)$ has a distributional derivative $u' \in L^2(0,T;H^{-1})$, does $(u^+)' \in L^2(0,T;H^{-1})$ make sense?

If $u \in L^2(0,T;H^1)$ has a distributional derivative $u' \in L^2(0,T;H^{-1})$, does $(u^+)' \in L^2(0,T;H^{-1})$ exist, i.e., is $u^+$ weakly differentiable in time? By $u^+$ I mean the positive ...
0
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1answer
35 views

Value of $\frac1{\Vert f\Vert}$

For $V=\{x\in X\mid f(x)=1\}$ show that $\inf\{\Vert x\Vert\mid x\in V\}=\frac1{\Vert f\Vert}$, where $X$ is banach and $f$ is a nontrivial element of the dual space of $X$. For $x\in V$ we have ...
0
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1answer
38 views

projection theorem for Banach spaces

In a remark to the projection theorem for Hilbert spaces I read this conjecture of a more general projection theorem: Let $X$ be a reflexive Banach space and $K\subset X$ closed and convex. Then for ...
2
votes
2answers
39 views

If $u_n \rightharpoonup u$ in $L^2(\Omega)$ and $u_n \geq 0$, is also $u \geq 0$?

If $\Omega$ is the usual bounded domain and $u_n \rightharpoonup u$ in $L^2(\Omega)$ and $u_n \geq 0$ a.e, is also $u \geq 0$ a.e? I know weak limits usually mess up things that one expects so I ...
0
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0answers
49 views

a completion of $c_{00}$ under some norm

Let $X$ be the completion of $c_{00}$ under some norm $\|\cdot\|_X$ satisfying $\|e_n\|_X=1$ for all $n$, where we let $(e_n)$ denote the canonical unit vectors in $c_{00}$. In Albiac/Kalton's book ...
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1answer
20 views

What other ideals are there in this subalgebra of the disk algebra

Let $A$ be the disk algebra and $A_0 = \{f \in A \mid f(0) = 0\}$. I am trying to give an example of a maximal non-modular ideal in $A_0$. I have tried $I=\{f\in A_0 \mid f(1) = 0\}$ and proved that ...
1
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2answers
20 views

Is this condition sufficient to ensure the locally convexity of a function at a given point?

Given $\bar x\in \mathbb R^n$. Let $f:\; \mathbb R^n\to \mathbb R$ be a nonconvex continuous function on $\mathbb R^n$ satisfying the followings (i) $f$ is not differentiable at $\bar x$, (ii) There ...
1
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1answer
31 views

If $u_n$ are equi-integrable and $f:\mathbb{R} \to \mathbb{R}$ is continuous, is $f(u_n)$ equi-integrable?

is it true that if $u_n$ are equi-integrable and $f:\mathbb{R} \to \mathbb{R}$ is continuous, is $f(u_n)$ equi-integrable? Here $u_n \in L^1(\Omega)$ where $\Omega$ is a finite measure space with ...
2
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2answers
46 views

What is the right domain for this Hamiltonian

I want to define a proper domain $D(H) \subset L^2$ for this Hamiltonian ( $\theta$, $\phi$ are the standard angles in spherical coordinates). Furthermore, the wave function is supposed to satisfy ...
2
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2answers
46 views

Weak convergence of partial sums

I recently came across an interesting problem on weak convergence in $\ell^2 (\Bbb N)$. Suppose that we have canonical basis $\{e_i\}$ in $\ell^2 (\Bbb N)$. We need to prove that the sequence ...
0
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0answers
25 views

calculate the sup of the max of 3 functions

Let a function be the variable, how the calculate the following expression? $$\inf_{c(t) \in C[-1,0]} \max \{ \max_{-1 \leq t \leq 0} |c(t)| , \max_{0 \leq t \leq 1} | \int_{0}^{t} c(v-1) +1 dv +c ...
1
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0answers
12 views

Existence of Galerkin approximations to PDE

Given a weak PDE $$ \langle b(u),v \rangle +\langle a(\nabla u), \nabla v\rangle=\langle f, v \rangle,\qquad v\in H^1_0 $$ where $b\in C(\mathbb{R})$ and $a\in C(\mathbb{R}^n,\mathbb{R}^n)$ grow ...
0
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0answers
28 views

Solution of nonlinear Schrödinger equation

Consider the linear Shr\"odinger equation, $$ (LS) \begin{cases} \partial_{t}u= i\Delta u, t\in \mathbb R,\\ u(x,0)=u_{0}(x), \end{cases} $$ $x\in \mathbb R^{n}.$ Taking the Fourier transform with ...
0
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2answers
24 views

Let $u_n \to u$ in $L^1(\Omega)$. Does $u_n^p \to u^p$ in $L^1(\Omega)$ if we know $u_n^p \in L^1(\Omega)$?

Suppose $u_n \to u$ in $L^1(\Omega)$ where $\Omega$ is a bounded domain. Suppose that $u_n^p \in L^1(\Omega)$ (actually $L^\infty(\Omega)$ for each $n$). Fix $p \in [1,\infty)$. So $u_n(x) \to u(x)$ ...
0
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0answers
13 views

Hermitian adjoint of an isometry

Let $u: H \to H$ be an isometric operator on a Hilbert space. Let $\ast$ be an involution. I was wondering if $u^\ast$ is also an isometry. I tried to prove it but didn't quite manage. Then I ...
0
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1answer
75 views

can anyone give me examples of open subspace of a metric space

Is there anyone who can give me an elegant example of non-empty subspace $A$ which is open in a metric vector space $H$? I know it cannot be found in $\mathbb R^n$..
1
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3answers
24 views

Lp space and sequence

For what value of $p$ the sequence $\displaystyle x_{n}=\frac{1}{n}$ is on $l^p$ (where $\displaystyle l^p = \lbrace (x_1,x_2,...)| x_{i}\in\mathbb{C}\hspace{0.1cm}\text{and}\hspace{0.1cm} ...
4
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0answers
42 views

a question about Tsirelson's space

Background. Let $T$ denote the Figiel-Johnson construction of the Tsirelson space, that is, the completion of $c_{00}$ under the implicitly-defined norm ...
3
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0answers
73 views

Solving 1D telegrapher's equation by reduction to two-dimensional wave equation

The solution $w : \mathbb R \times \mathbb R_{+} \to \mathbb R$ of the Cauchy problem for the telegrapher's equation $$ w_{tt} - c^2 w_{xx} + c^2 \lambda^2 w = 0 $$ with $$ w(x,0) = 0, \qquad ...
0
votes
1answer
21 views

If $u \in L^2(0,T;H^1)$ with $u_t \in L^2(0,T;H^1)$ and $u$ and $u_t$ are bounded, is $u \in C([0,T];L^\infty$)?

Let $u \in L^2(0,T;H^1(\Omega))$ with $u_t \in L^2(0,T;H^{-1}(\Omega))$ so $$\int_0^T u(t)\varphi'(t) = -\int_0^T u'(t)\varphi(t)$$ holds for all $\varphi \in C_c^\infty(0,T)$. Suppose we know ...
2
votes
2answers
26 views

Bounded, surjective linear operator between Banach spaces

How can I show that for a given surjective linear operator $T: X \to Y$ between Banach spaces, if there exists an $\epsilon > 0$ such that $||Tx|| \geq \epsilon||x||$ for all $x \in X$, then $T$ is ...
1
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1answer
56 views

Kaplansky's Density Theorem for Unitary Operators

Let $M \subseteq B(H)$ be a *-subalgebra containing the identity on H. If there is a unitary T in the unit ball of the SOT-closure of $M$, is there a net of unitary oprators in the unit ball of $M$ ...
1
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1answer
14 views

Approximate continuous function that vanishes at origin by odd powers polynomial

Prove or disprove: for every real-valued continuous function $f$ on $[0,1]$ such that $f(0)=0$ and every $\epsilon $, there is a polynomial $p$ having only odd powers of $x$, i.e., $p$ is of the form ...
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0answers
21 views

the best approximate element

$X$ denotes the set $\{ f \in C[-1,1] | f ~ is~ continuous~ differentiable~ on~ [0,1]\}$, $Y$ denotes $\{ f \in X | f ~satisfies~ $f '(t) = f(t-1)$~ on~ [0,1]\}$. For any $f\in X$, does there ...
2
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0answers
16 views

When a locally convex space is metrizable

Let $X$ be a locally convex Hausdorff topological space, whose topology if generated by the countable family of seminorms $\{p_i:\space i\in\mathbb{N}\}$. I'd like to prove that $X$ is metrizable. So, ...
0
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1answer
31 views

If $u_n \rightharpoonup u$ in $L^2(\Omega)$, does $u_n^+ \rightharpoonup u^+$ in $L^2(\Omega)$?

Let $\Omega$ be a bounded domain. If $u_n \rightharpoonup u$ in $L^2(\Omega)$, does $u_n^+ \rightharpoonup u^+$ in $L^2(\Omega)$ where $u_n^+ = \max(0,u_n)$. Note all convergences are weak. My ...
3
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1answer
57 views

Characterization of weak solution

5 Nonlinear elliptic variational inequalities Preliminaries In order to explain the importance of elliptic variational inequalities, first consider the weak solution of the linear ...
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1answer
42 views

Elegant way to solve this extreme value problem

I want to show that $$ \sup_{(x,y)\in \mathbb{R}^2 \setminus \lbrace (0,0) \rbrace} \frac{(ax+by)^2}{x^2+y^2} =a^2+b^2 $$ where $a,b \in \mathbb{R}$ are fixed (this problem appears when one tries to ...
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0answers
18 views

Reference for Hölder space $C^{k,\beta}(X,Y)$, X and Y Banach spaces

Does anybody know of a reference for the Hölder spaces $C^{k,\beta}(X,Y)$, which treats the case where $X$ and $Y$ are (subsets of) Banach spaces? (Or something more general.) All books I have seen ...
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0answers
20 views

Argument missing..

claim: X complete metric space $\Rightarrow$ X not meagre Proof: Assume X meagre$\Rightarrow X=\bigcup_{n\geq1} A_n, ($closure$(A_n))^°\neq \emptyset\forall n\Rightarrow X=\bigcup_{n\geq ...
1
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1answer
29 views

A function sequence converge to a Fourier series implies point-wise converge?

Assume $f(x)$ is a smooth $2\pi$ periodic function and can be decomposed as $$f(x)=\sum_{n=-\infty}^\infty a_ne^{inx}$$ A function sequence $f_m(x)$ satisfies $$\int_0^{2\pi}f_m(x)e^{-inx}dx\to ...
1
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2answers
32 views

Are bounded linear functionals on $L^{\infty}$ of “bounded variation?”

Let $(X,\mathscr M,\mu)$ be a measure space and let $L^{\infty}$ be the set of (equivalence classes of) essentially bounded measurable functions on it. Suppose that $\phi\in (L^{\infty})^*$; that is, ...
0
votes
1answer
28 views

Using Gauss's Theorem in weak formulation

Can anyone see how exactly Gauss's theorem used in the following case: In defining the weak solution to the linear elliptic equation we start with $$-\sum_{j,k=1}^{n}D_{j}(a_{jk}D_{k}u) + cu = f ...
1
vote
1answer
31 views

Let $u_n, u \in L^2$. If $\int u_nv \to \int uv$ for all $v \in H^1$, does $\int_{}u_nh \to \int_{}uh$ for all $h \in L^2$?

Let $\Omega$ be a bounded domain and let $u_n$ and $u \in L^2(\Omega)$. Question: If $\int_{\Omega}u_nv \to \int_{\Omega}uv$ for all $v \in H^1(\Omega)$, does $\int_{\Omega}u_nh \to \int_{\Omega}uh$ ...
1
vote
1answer
76 views

approximating continuous function on $[0,1]$ by monotone increasing polynomials

Let $f\in C[0,1]$ be real-valued. Prove that there is monotone increasing sequence of polynomials $\{p_n(x)\}^\infty_{n=1}$ converging uniformly on $[0,1]$ to $f(x)$. Yea, it should be done by ...
1
vote
1answer
23 views

$g(T)$ bounded implies $T$ bounded, if $T$ is linear and $g$ is bounded linear functional

Suppose $X$ is a Banach space and $y$ is a normed linear space and $T :X\to Y$ is linear map such that for every bounded linear functional $g\in Y^*$ we have $g( T)$ is bounded . Show that $T$ is ...