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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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
46 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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17 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
7 views

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
20 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
23 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
28 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
15 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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16 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
9 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
14 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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1answer
14 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
22 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
28 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] ...
3
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1answer
24 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
40 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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1answer
21 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
35 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
15 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
27 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
12 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
14 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
22 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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21 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
11 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
10 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
29 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
20 views

Banach Spaces: Improper Riemann Integral

Disclaimer This thread is related to: Stone's Theorem Definition Given a measure space $\Omega$ and a Banach space $E$. Consider functions $F:\Omega\to E$. Denote the measurable subsets of finite ...
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1answer
45 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
26 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 ...
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1answer
49 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 ...
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1answer
25 views

Approximation of $f \in L^1_{loc}$

I am trying to prove the following statement: If $\Omega$ open in $\mathbb{R}^n$, $f \in L^1_{loc}(\Omega)$ (a set of all functions whose integrals on compact sets exist) and $\int_{\Omega}f\cdot g ...
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0answers
9 views

Visual notion of tangential gradient

Before I begin, this question is related to personal reading and is not in any way connected to an assessment/assignment. I am struggling to visualise the tangential gradient. As I understand it, the ...
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1answer
20 views

Find a tight frame of exponentials for $L^2(T)$, where $T \subset \mathbb{R}^2$ is a triangle with vertices $(0,1)$, $(1,0)$, and $(-1,0)$.

Find a tight frame of exponentials for $L^2(T)$, where $T \subset \mathbb{R}^2$ is a triangle with vertices $(0,1)$, $(1,0)$, and $(-1,0)$. Normally, I would do is find a matrix representation and ...
2
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1answer
35 views

Verify that the set $\{ (b-a)^{-\frac{1}{2}}e^\frac{2\pi ixj}{b-a} \}$ is an orthonormal basis of $L^2(a,b)$.

Let $H=L^2(a,b)$ with $a<b$. Verify that the set $\{ (b-a)^{-\frac{1}{2}}e^\frac{2\pi ixj}{b-a} \}$ is an orthonormal basis of $L^2(a,b)$. Verify also that $$\{(b_1-a_1)^{-\frac{1}{2}} \cdot \dots ...
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2answers
23 views

Norm of orthogonal projection operator $P$, if $\text{Im}\,P\subseteq\text{Im}\,Q$, with $Q$ also an orthogonal projection

In Rynne & Youngson: Linear Functional Analysis, there is an exercise stated as Let $\mathcal{H}$ be a complex Hilbert space and let $P$, $Q\in B(\mathcal{H})$ be orthogonal projections. Show ...
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0answers
24 views

$[f=q]$ is closed implies $f$ is continuous. [on hold]

Define $H=[f=q]$ to be the set of points such that $f(x)=q$. Let us assume that H is closed. Then the complement of H is open and nonempty (since $f$ does not vanish identically - ""i don't understand ...
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2answers
30 views

Weak topology : defined by linear mapping vs semi-norm

Wikipédia: -The weak topology on X is the initial topology with respect to X* (let's note it T') -If the field K has an absolute value , then the weak topology σ(X,F) is induced by the family of ...
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2answers
30 views

Book for Hilbert spaces.

Which book either on functional analysis or specifically for Hilbert spaces has the best way of explaining with most examples and to the point without much applications. I studied Limaye's book and ...
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1answer
31 views

Determine adjoint operator

Let $K\in D\{(x,\xi)\in \mathbb {R}^2 : x > 0, \xi > 0\} $ and $L (\phi)(x)=\int_0^x K (x,\xi)\phi (\xi)d\xi $ for $\phi \in D (\mathbb {R})$ $D $ is the space of testfunctions I know that ...
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1answer
32 views

How prove this indentity$\langle F'',g\rangle=-\frac{1}{4}\int_{0}^{+\infty}\frac{g(x)-g(0)}{x^{3/2}}dx$

For the generalized function F defined as $\langle F,g\rangle =\int_{0}^{\infty}\sqrt{x}g(x)dx$,show this the following equalities $$\langle F'',g\rangle ...
4
votes
1answer
48 views

Fourier transform inversion formula for $f\in L_1(\mathbb{R}^n)$ and Dini condition

Let us define the Dini condition for a function $f\in L_1(-\infty,\infty)$, i.e. Lebesgue summable on $\mathbb{R}$, as Given an $x\in\mathbb{R}$ there is a $\delta>0$ such that the Lebesgue ...
0
votes
1answer
71 views

Is this equality true or it is not necessarily true?

Let $A$ and $B$ are two factor von neumann algebras that act on two infinite dimensional Hilbert spaces H and K respectively. Let $\Phi:A\longrightarrow B$ is an additive bijective map with some other ...
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2answers
39 views

Show $u\in H^1(B(0;1/2))$ is holder continuous, where $u$ is a weak solution to $-\Delta u+cu=f$ for some $c\in L^q$ for some $3/2<q<2,$.

If $u\in H^1(B)$, $B=\lbrace x\in\mathbb{R}^3, |x|<1/2\rbrace$ is a weak solution to $$-\Delta u+cu=f$$ for some $c\in L^q$ for some $3/2<q<2,$ and $f\in C^\infty$, then show $u$ is holder ...
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0answers
26 views

Subset being orthonormal basis if $\sum_{n\geq 1} \|a_{n}-b_{n}\|^{2} < \infty$ [on hold]

I have a question Supposing $X$ is a Hilbert space and lets suppose that $A=\{a_{n} \ : \ n\geq 1\}$ is an orthonormal basis of $X$. Let $B=\{b_{n} \ : \ n\geq 1 \}$ be an orthonormal subset of $X$. ...
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0answers
28 views

Is there are “sphere” associated to any topological vector space?

If I have a topological vector space that is not locally compact, is it still possible to associate to it some natural "sphere" like object? For locally compact Hausdorff spaces, the my first guess ...