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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Hardy-Littelwood-Sobolev without Marcinckiewicz?

Here is the statement of the Hardy-Littlewood-Sobolev theorem. Let $0< \alpha< n$, $1 < p < q < \infty$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$. Then: $$ \left \| ...
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2answers
28 views

Strongly convergent to zero in $L^2$ but $H^1$ norm not vanishing

Let $\Omega$ be some open, bounded, smooth subset of $\mathbb{R}^n$. I'm wondering whether it is possible for a sequence of functions $f_n:\Omega \rightarrow \mathbb{R} $ to be strongly convergent to ...
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0answers
19 views

Weak-* convergence in Sobolev spaces

Let's consider a sequence $\{f_n\}_n$ in the space $L^\infty(0,T;H^1(\mathbb{R}^n))$. What does it mean that $\{f_n\}_n$ converges weakly-* in $L^\infty(0,T;H^1(\mathbb{R}^n))$?
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32 views

The space $C[0,1]$ with the metric defined by $d(f,g)=\int _0^1|f(t)-g(t)|dt$?

how to show this set is separable ? The space $C[0,1]$ with the metric defined by $d(f,g)=\int _0^1|f(t)-g(t)|dt$?
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1answer
28 views

Closed map on Banach Space

Let $X,Y$ be Banach spaces and $T \in B(X,Y)$. Show that if $T$ sends every bounded closed subsets of $X$ onto closed sets of $Y$ then $T(X)$ is closed. It's true when the map is injective, but is it ...
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1answer
24 views

Show that any bounded linear functional on a normed linear space is continuous

Show that any bounded linear functional on a normed linear space is continuous. Can we say that it is uniformly continuous ? Also, is it true, if we reverse the statement any continuous linear ...
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2answers
39 views

If $\lim_{x\to\infty} [f(x+1)-f(x)] =l$ then $\lim_{ x\to\infty}f(x)/x =l$ ($f$ is continuous)

Prove that if $f$ is continuous on $\mathbb R$ and $$\lim_{x \to +\infty} [f(x+1)-f(x)] = l,$$ then $$\lim_{x\to +\infty} f(x)/x =l.$$ So I've been trying for hours to use the series ...
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0answers
22 views

Bounded Measurable Functions: Pointwise Limit vs. Uniform Limit

Agreement All notions are up to null sets. Limits are meant by simple functions. Problem Given a finite measure space $\mu(\Omega)<\infty$ and a Banach space $E$. Consider bounded measurable ...
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1answer
11 views

Generalized Riemann Integral: Uniform Convergence

Disclaimer This thread is meant to record. See: Answer own Question And it is written as question. Have fun! :) Reference This thread is related to: Generalized Riemann Integral: Nonexample? ...
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0answers
13 views

Projections as a dense subset of $\ell_p^*$ [on hold]

I'm trying to prove that weak convergence in $ \ell_p^*$ is equivalent to coordinatewise convergence and boundedness, i.e. $$ a_n^{(k)} \text{ converges weakly to}~ a_0^{(k)} \iff \exists_{ M>0} ...
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1answer
12 views

Equivalence of dual spaces of Sobolev Spaces

I have a quick question: Is the following equivalence true for Sobolev Spaces $(W^{1,p}(\Omega))^{*} = W^{-1,p}(\Omega) = (W^{1,p}_{0}(\Omega))^{*}$ where $W^{1,p}_{0}(\Omega)$ is the closure of ...
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1answer
9 views

norm of bounded linear operator restricted to dense subspace

I have no idea how to do this question I was given in class. Let $E$ and $F$ be normed spaces and let $T \in \mathcal{L}(E,F)$. Suppose that $E_0 \subseteq E$ is a dense subspace. Show that ...
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1answer
21 views

Under what conditions, depending on my particular choice of $f$ and $g$ (other than $f\in L^1$ and $g\in L^\infty$) under which $f\cdot g$ is in L^1?

Is there a possibility of a function $f\in L^{1}(\Omega)$ and $g \in L^{2}(\Omega)$ implying that $f\cdot g\in L^{1}(\Omega)$? At least can you let me know what suitable function space $g$ is in some ...
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1answer
46 views

A Challenge on linear functional and bounding property

I took a midterm exam and after that wrote this problem down. My instructor was unable to solve it. The problem is copied here in order for anyone to help me. Suppose $f:E\to \mathbb{C}$ is a ...
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1answer
13 views

Uniform Boundedness Principle

This is a homework question, I have no clue where to start! Use Uniform Boundedness Principle to prove the following statement. A subset $X$ of a normed space $E$ is bounded if and only if $f(X)$ is ...
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0answers
22 views

Unital maps taking values in abelian C*-algebras

It is known that a bounded linear functional $f$ on a unital C*-algebra $A$ is positive if and only if $f(I)\geqslant 0$. Is the same true for bounded linear operators $T\colon A\to C(X)$ with $T(I) = ...
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3answers
55 views

An example of a function in $L^1[0,1]$ which is not in $L^p[0,1]$ for any $p>1$

Title says most of it. Could you help me find an example? It is easy obviously to show a function that would not be in $L^p[0,1]$ for a specific $p$ (say $(1/x)^{1/p}$, but I can't see how it would ...
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1answer
18 views

Showing self adjointness

$\pi:$ $Lx=\sum_{j=0}^{n}(p_{n-j}x^{(j)})^{(j)}$,$\,\,$ $x^{(j)}(a)=x^{(j)}(b)=0,\, j=0,1,...,n-1.$ where $p_{n-j}\in C^{n-j}[a,b]$ are real and $p_0(t)\neq0$ on $[a,b]$. I want to show that the ...
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0answers
21 views

Prove domain of Dependence Inequality for the Wave Equation?

Let $(x_0,t_0)\in R^{n+1}$ with $t_0>0$, and let $\Omega$ be the conical domain in $R^{n+1}$ bounded by the backward characteristic cone with apex at $(x_0,t_0)$ and by the plane $t=0$. Suppose ...
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1answer
25 views

If $f \in L^{1}(\Omega)$, $g \in L^{2}(\Omega)$, with $\Omega$ a bounded domain in $\mathbb{R}^n$, then can $f.g$ be in $L^1{\Omega}$?.

I have come up with an argument which is as follows. Please correct me if it makes no sense. Consider $\int_{\Omega}|f.g|dm$. Then if $|f|\in L^1$ then $\sqrt{|f|}\in L^2$. Hence we have ...
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2answers
28 views

Stone's Theorem and Functional Calculus

I've asked a few questions on here before regarding functional calculus but I am still having a bit of trouble. I have been reading up on Stone's theorem for unitary groups, and going through the ...
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1answer
34 views

Reproducing kernel Hilbert space, why?

Let $K: X \times X \rightarrow \mathbb{C}$ be a positive definite kernel on a set $X$, i.e. for any $x_1, \cdots, x_n \in X$, the matrix $$ [K(x_i, x_j)]_{ij} \in \mathbb{C}^{n \times n} $$ is ...
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1answer
17 views

How do I construct a Schauder basis of $l^2$ with the set $\{v_n\}

How do I construct a Schauder basis of $l^2$ with the set $\{v_n\} \subset l^2$, where $v_n=\frac{1}{\sqrt{2}}(e_n-e_{n+1})$ if $n$ is odd and $v_n=\frac{1}{\sqrt{2}}(e_n+e_{n-1})$ if $n$ even. Note ...
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0answers
11 views

$L^p$ is a quasi normed space for $0<p<1$ [duplicate]

I know that $L^p$ is a vector space for $p>0$ and a normed space for $p \geqslant 1$ now I need show that for $ 0<p<1$ and $f,g \in L^p$ exist $K \in \mathbb{R}$ such that $||f+g||_p ...
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0answers
20 views

Choosing a clever “test function” in Sobolev spaces.

Given $\mathbf{f}$ with $f_1,...,f_N\in L^2(\Omega)$ $$\int_\Omega \mathbf{f} \cdot \nabla v = 0 \quad\forall v \in H_0^1(\Omega)$$ we have $\mathbf{f} = \mathbf{0}$ a.e. since $\mathbf{f} \in ...
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0answers
10 views

Wiener's lemma and Hulanicki's lemma

Let $\mathcal{A}(\mathbf{T})$ be the Banach algebra of continuous complex-valued functions on the unit circle with absolutely convergent Fourier series. Then Wiener's lemma states that if $f \in ...
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0answers
19 views

Weak convergence in $l^p$-space [on hold]

Let $1<p<\infty$, $x_n=(x_n^{(j)})_j\in l^p$ for $n\in \mathbb N$ and $x=(x^{(j)})_j\in l^p$. Show that $$x_n\rightharpoonup x\iff \forall j\in \mathbb N:x_n^{(j)}\rightarrow x^{(j)}, ...
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0answers
20 views

Weak convergence in Hilbert space implies strong convergence of averages for some subsequence

Let $H$ Hilbert Space. Show that if $x_n\rightharpoonup x$ then there exists a subsequence $\{x_{nk}\}$ of $\{x_{n}\}$ such that the sequence $\lim_{m\rightarrow \infty } ...
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1answer
25 views

an additive bijective map

Let H be a Hilbert space and $\Phi:B(H)\longrightarrow B(H)$ is an additive bijective map. If $\mathbb{R}I⊆\Phi(\mathbb{R}I)$, can we conclude by the bijectivity of $\Phi$ that ? ...
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1answer
32 views

Projection of $H^1([0,1])$ on its subspace .

Let $H^1([0,1])$ be the Sobolev space $W^{1,2}([0,1])$ with the scalar product $\langle f,g\rangle = \int_0^1 fg + \int_0^1 f'g'$. We can consider the closed and convex subset $K=\{f \in H^1([0,1] ...
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1answer
26 views

Graph of weakly continuous linear operator

I have a few questions regarding the graph of an operator. Consider the operator $T:X \rightarrow Y$ between Banach spaces $X,Y$. Assume that $T$ is a linear operator which is (weak, weak)-continuous, ...
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2answers
45 views

$\int_{-1}^{1}|f(t)|dt \geq C\left(\int_{0}^{2}|f(t)|^2\right)^{1/2}$ for polynomials

Prove that there exists constant $C>0$ that for all $f \in P_n$ we have: $$\int_{-1}^{1}|f(t)|dt \geq C\left(\int_{0}^{2}|f(t)|^2\right)^{1/2}$$ Where $P_n$ is space of polynomials with ...
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votes
1answer
22 views

Find eigenspectrum of $T^{*}$

Let {$u_n$} be an orthonormal basis of Hilbert space $H$. Let $T \in B(H)$ s.t. $T(u_n)=u_{n+1}$ for $n \ge 1$. Find eigenspectrum of $T^{*}$. I have tried to find it taking the corresponding matrix ...
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1answer
49 views

Hermite functions as eigenvectors of Fourier transform

In order to find an orthogonal basis of eigenvectors of the Fourier transform operator $F:L_2(\mathbb{R})\to L_2(\mathbb{R})$, $f\mapsto\lim_{N\to\infty}\int_{[-N,N]}f(x)e^{-i\lambda x}d\mu_x$ for ...
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1answer
16 views

Show $\sigma(T)=\sigma{(\overline{T^{*}})}$

Let $T \in B(H)$ be a bounded operator. Is $\sigma(T)=\sigma{(\overline{T^{*}})}$ true for $T$? $\textbf{TRY-}$ I have proved it is true for normal operator but could not do it for bounded ...
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0answers
28 views

Show that both $S$ and $T$ are bounded. [duplicate]

Let $S$ and $T$ be linear map defined on hilbert space $H$ s.t. $\langle Su,v \rangle=\langle u,Tv\rangle $ $\forall u,v \in H$. Then show that both $S$ and $T$ are bounded. $\textbf{TRY-}$ If I can ...
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0answers
15 views

When is a metrizable topological vector space locally bounded?

Consider a topological vector space $E$ with topology $\sigma$. Suppose that $E$ is metrizable, in other words, that there exists a metric $d$ on $E$ that induces the topology $\sigma$. One can then ...
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1answer
16 views

find a sequence converging to zero but not the elemet of lp space for every 1<_p<infinity

I am studying functional analysis and I have a problem about finding a sequence converging to zero such that this sequence is not in lp for every p. By lp I mean lp={(x_k)=(x1,x2,...):Σ|x_k|^p_2 it is ...
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3answers
23 views

When talking about a normed vector space, does it's metric always need to be the induced one?

The title basically says it all. If we have a normed vector space, is it possible to work with the space as a metric space with a different metric than the induced one? So if the space is $(X,||\ ...
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0answers
25 views

The space of distribution $H^{-1}$

Let's suppose to have a function $u$ in the space $L^\infty(0,T;H^1(\mathbb{R}^n))$ with $\partial_t u\in L^\infty(0,T;H^{-1}(\mathbb{R}^n))$. So $\partial_t u$ is a linear and continuous functional ...
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1answer
12 views

Positive invertable element of a C*- algebra

The following is Theorem 2.2.5 of Murphy's C*-algebras and operator theory: Let $A$ be an unital C*-algebra and $a,b$ are positive invertable elements, if $a\leq b$, then $0\leq b^{-1}\leq a^{-1}$. ...
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1answer
11 views

A minimisation problem in an Hausdorff space

Let X an hausdorff space, K a compact set, f a lower continuous function. How can you prove that f has a minimum on K? Ps: be careful X is not specially a metric space
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1answer
30 views

As $\lambda \to \infty$, $||R_{\lambda}(T)|| \to 0$

"Let $T \in B(X,X)$. Prove that $||R_{\lambda}(T)|| \to 0$ as $\lambda \to \infty$." We have that $R_{\lambda}(T)x=(T-\lambda I)^{-1}(x)$. The problem doesn't specify but the book says that they will ...
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0answers
26 views

Banach Spaces: Improper Riemann Integral

Reference This thread is related to: Stone's Theorem For a bounded example of non-integrability see: Riemann Integral: Nonexample? For a comparison of integrals see: Uniform Integral vs. Riemann ...
5
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1answer
46 views

Theorem 3.54 (about certain rearrangements of a conditionally convergent series) in Baby Rudin: A couple of questions about the proof

Here's the statement of Theorem 3.54 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Let $\sum a_n$ be a series of real numbers which converges, but not absolutely. Suppose ...
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1answer
14 views

Topology induced by semi-norm and P-topology

Let V be an abstract vector space over F and P a family of semi-norms on V. We can define a topology on V by declaring its base to consist of subsets of V of the form {v∈V | p1(v−x)<ϵ ∧ … ∧ ...
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1answer
30 views

Proving dense set is core for a self adjoint operator

Let $A$ be a self adjoint operator in a Hilbert space $H$ and $D\subseteq D(A)$ a dense subset such that $$ e^{iAt}:D \to D. $$ How can I show that $D$ is a core for $A$? I need to show that ...
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0answers
25 views

a problem about the application of Banach-Steinhaus theorem [on hold]

Let $\{x_k\}$ to be a sequence in Banach space $X$. Prove that if for any f in $X^*$, $\sum_{k=1}^\infty |f(x_k)|<+\infty$, then there exist a constant M s.t. for every $f\in X^*$ we have ...
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0answers
15 views

Green-Operator for Sturm-Liouville Differential equation compact on Sobolev space?

Let $g$ be Green's Function for a Sturm-Liouville differential equation. Is the operator $G: H_{0}^{1}(0,1) \rightarrow H_{0}^{1}(0,1)$ defined by $(Gf)(x) := \int_{0}^{1} g(x,y)f(y) dy, \quad f \in ...
1
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1answer
52 views

Recapitulated: Stone's Theorem Integral

This problem grew out from: Stone's Theorem Integral For a definition, a nonexample and a comparison see: Generalized Riemann Integral: Definition Generalized Riemann Integral: Nonexample ...