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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2
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2answers
41 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 ...
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0answers
51 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 ...
0
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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
vote
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
votes
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
votes
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
vote
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 ...
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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
votes
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
votes
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
votes
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
vote
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
votes
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
vote
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
vote
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 ...
0
votes
0answers
22 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
votes
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
votes
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
votes
1answer
58 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 ...
1
vote
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 ...
0
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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 ...
0
votes
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
vote
1answer
30 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
vote
2answers
33 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
24 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 ...
0
votes
1answer
53 views

Balanced Core: $U\text{ open }\implies U^*\text{ open}$

I need one last lemma for the proof of finite dimensional subspaces are closed: Is it true that if a subset is open so is its balanced core??
5
votes
0answers
99 views

Connections and dependences between topological and algebraic basis in topological vector space

On my last functional analysis exam, one of the tasks was to show that if normed vector space $X$ have countable Hamel basis, then $X$ is separable space (over field $\mathbb{R}$). I am not sure if ...
4
votes
1answer
48 views

*-representations of dense subalgebras

Let $H$ be a separable Hilbert space and let $K(H)$ be the C*-algebra of compact operators on $H$. Suppose that $A$ is a *-subalgebra of $K(H)$ which contains all the finite-rank operators. Given a ...
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votes
0answers
37 views

Green function of Sturm liouville problem

How to find the Green function of the following problem: $$\begin{cases}-(p(t)u')'+q(t)u=f(t,u), t>0\\u(0)=u(+\infty)=0\end{cases}$$ where $\displaystyle\frac{1}{p},\frac{1}{q}\in ...
1
vote
0answers
20 views

functions with several variables: show injectivity

Let $a,b \in \mathbb R,~ 0 < a < b$. Let $f=(f_1,f_2,f_3): \mathbb R^2 \rightarrow \mathbb R^3$ be defined by: $f_1(s,t) = (b+a \cdot cos(s))cos(t)\\f_2(s,t)=(b+a \cdot ...
0
votes
0answers
13 views

Statement of Markov-Kakutani fixed-point theorem

Markov-Kakutani fixed-point theorem is usually stated as follows: "Let $E$ be a locally convex topological vector space. Let $C$ be a compact convex subset of $E$. Let $S$ be a commuting family of ...
3
votes
2answers
50 views

What is $T^nf(t)$? (Question on integrals)

I am supposed to prove the following: For the operator $T$ defined by $$Tf(t)=\int_0^t(t-s)f(s)\,ds,\quad f\in C[0,1]$$ Show that $$T^nf(t)=\int_0^t\frac{(t-s)^{2n-1}}{(2n-1)!}f(s)\, ds$$ I ...
0
votes
0answers
29 views

Measure theory for $L^2$

I have some questions about measure theory. $u,v,w \in L^{2}(\mathbb{R})$ We suppose $u,v,w$ are bounded and $|u(x)|\leq |v(x)|+|w(x)|$, $|u(x)-u(y)|\leq |v(x)-v(y)|+|w(x)-w(y)|$ for all $x,y \in ...
0
votes
1answer
45 views

Clarification for this exercise needed

I would like to solve the following exercise but there are a few minor things I am not clear about: Let $A$ be the Banach algebra of $C^1([0,1])$ endowed with the norm $\|f\|=\|f\|_\infty + ...
4
votes
1answer
115 views

If $u_n \to u$ in $C([0,T];H^{-1})$ and $\lVert u_n\rVert_{L^\infty} \leq C$ then $u_n(t) \rightharpoonup u(t)$ in $L^1(\Omega)$?

Let $u_n \to u$ strongly in $C([0,T];H^{-1}(\Omega))$ and suppose that $u_n$ is uniformly bounded in $L^\infty((0,T)\times\Omega)$. Then $$u_n(t) \to u(t)$$ weakly in $L^1(\Omega)$ for each $t$? Can ...
0
votes
1answer
19 views

Question about a scalar product

It is well known that in a hilbert space $H$ with orthonormal basis $(e_n)_{n=1}^{\infty }$, we have for every $f, g \in H$ $$\displaystyle{\langle f,g\rangle=\sum_{n\in\mathbb{N}}\overline{ \langle ...
0
votes
0answers
15 views

Bessel-like inequality

Let $\{e_n\}$ be an orthonormal sequence in an inner product space E. Then I'm trying to show the following inequality: $$\sum_1^\infty| \langle x, e_n \rangle \langle y, e_n \rangle | \leq ...
0
votes
1answer
36 views

Topological Vector Space: Uniform Structure

Disclaimer: This thread is meant informative and therefore written in Q&A style. Of course everybody is encouraged to give an answer as well! Prove that any topological vector space gives rise ...
1
vote
1answer
43 views

Space of Distribution wrt to topology of uniformly convergence on bounded sets not Frechet-Space.

I found a state, that the Space of Distribution on (here:) $\mathbb{R}^{n}$, which is equipped with the topology of uniformly convergence on bounded subsets is not a Frechet Space. As far as i can ...
0
votes
1answer
31 views

Continuous functions on a compact Hausdorff space

Are there any nice properties that characterize the space $C(K)$, where $K$ is compact Hausdorff? I mean results on the whole space (not each function in particular), for example, is it true that ...
1
vote
1answer
24 views

If $f:\mathbb R \to \mathbb R$ is continuous and piecewise $C^1$ with $f'$ bounded and $u \in L^2(0,T;L^2)$ then $f(u) \in L^2(0,T;L^2)$

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain. If $f:\mathbb R \to \mathbb R$ is continuous and piecewise $C^1$ with $f'$ bounded, and if $u \in L^2(0,T;L^2(\Omega))$ then $f(u) \in ...
2
votes
1answer
20 views

Baire Category for monotonic sequence

Let X be a non-empty complete metric space and let $\{f_n:X\to \Bbb R\}^\infty_{n=1}$ be a sequence of continuous functions with the following property: for each $c\in X$, there exists an integer ...