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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Evaluating 'Constant' Term

suppose I have a pde $$u_{xt}(x,t)+u(x,t)u_{xx}(x,t)=h(t),\,\,\,\, x\in[0,\pi],\,\, t>0$$ for some unspecified function $h(t)$. This question is about finding what $h(t)$ is. Please, you may ...
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27 views

Generalized Fourier series in $L^2$ that do not converge pointwise a.e.

For a Hilbert space $L^2$ we have the notion of an orthonormal basis $\{f_j\}$ being a sequence of orthonormal elements such that any element $f$ in $L^2$ can be approximated by partial sums in terms ...
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0answers
21 views

Direct Sum: Stone

Problem Given Hilbert spaces $\mathcal{H}_\alpha$. Consider Hamiltonians: $$H_\alpha:\mathcal{D}H_\alpha\subseteq\mathcal{H}_\alpha\to\mathcal{H}_\alpha:\quad H_\alpha=H_\alpha^*$$ And their ...
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1answer
13 views

Is the identity in unital, simple, purely infinite $C^*$-algebra always infinite?

I'm trying to prove that the identity, $I$, of a unital, simple, purely infinite $C^*$-algebra is always an infinite projection. What I'm hoping is that the following is true: If $p$ in ...
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0answers
40 views

A doubt regarding derivative of convolution!!

In the following calculation: $\int_{\mathbb R^{d}} u_{o \epsilon} div (\phi) dx = \int_{\mathbb R^{d}} (u_{o} * \psi_{\epsilon}) div(\phi) dx = \sum_{i=1}^{d} \int_{\mathbb R^{d}} ( u_{o} * ...
3
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3answers
49 views

Which Hilbert space is isometrically isomorphism with $B(E)$ for some Banach space $E$.

Consider $H$ as a Hilbert space. How can I find a Banach space $E$, for that, $H=B(E)$ where $B(E)$ is the set of bounded linear operator on $E$? (At least under some conditions on $H$) Also if the ...
1
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1answer
31 views

Square root of a compact normal operator

Halmos expresses below problem in his book; Problem: If $A$ is a normal operator and if $A^n$ is compact for some positive integer $n$, then $A$ is compact. I have an example in my mind which I ...
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2answers
72 views

Is every Hilbert space a Banach algebra?

Let $H$ be a Hilbert space. Could we say that, always there is a multiplication on $H$, that makes it into a Banach algebra? If not, under which conditions does it exist?
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1answer
15 views

How to prove this “local invertibility” theorem for bounded linear operators?

The theorem states that, suppose $X,Y$ are complete normed vector spaces, if $\mathscr A_0\in \mathscr L(X;Y)$ is invertible (i.e., $\exists \mathscr A_0^{-1}\in\mathscr L(Y;X)$ s.t. $(\mathscr ...
3
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1answer
32 views

The same topologies

Let $L^1 (\mathbb{Z})$ be the space of all functions $f:\mathbb{Z}\rightarrow \mathbb{C}$ such that $\left\{\|f\|=\sum_{k\in \mathbb{Z}}|f(k)|<\infty\right\}$. Clearly, $L^1 (\mathbb{Z})$ is a ...
2
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0answers
17 views

Reference request: The compactness and compact embedding in Besov Space?

This post has been on MathOverflow for couple of days but receive no response. So I put it here hoping for more attentions. Thank you guys! Let $\Omega\subset \mathbb R^N$ be open bounded with ...
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0answers
13 views

Gaussian unitary dilation of Gaussian channels

I am starting with a few definitions. All these are standard and can be accessed from some quantum information or quantum physics books, for instance the books by Holevo or Parthasarathy. The question ...
3
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1answer
45 views

A question on the Banach fixed point theorem.

Suppose $f:(X,\tilde{d})\rightarrow(X,d)$ be a continuous function satisfying \begin{eqnarray}d(f(x),f(y))\leq \lambda d(x,y),\end{eqnarray} $\lambda > 1$. Let $\tilde{d}(x,y)=\lambda d(x,y)$. I ...
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1answer
50 views

How to prove the following inequality? (or a counter example)

We know that we have $[\int |f(x)|^{p} \mu(dx)]^{1/p}\leq [\int |f(x)|^{q} \mu(dx)]^{1/q}$ when $p\leq q$, where $\mu$ is a probability measure and $f$ is a smooth function. Do we in general have the ...
2
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1answer
36 views

Boundedness of an operator with kernel

Let $K(x,y)$ be measurable in $\mathbb{R}^2.$ Suppose there is a positive, measurable (w.r.t Lebesgue measure on $\mathbb{R}$) $w(x)$ and $A\geq 0$ such that $$\int_{-\infty}^\infty \vert K(x,y) ...
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1answer
59 views

Trace: Independence

Problem Given a Hilbert space $\mathcal{H}$. Consider an operator: $$A\in\mathcal{B}(\mathcal{H}):\quad\operatorname{Tr}|A|<\infty$$ Regard ONB's: ...
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2answers
52 views

Deny Lions Lemma

I am working through the finite element book by Ciarlet and am currently looking at the Deny Lion's Lemma (Theorem 3.1.1 p. 115). The Lemma essentially wants to show that $\inf_{p \in P_{k}}\Vert v ...
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2answers
27 views

Prove that orthonormalsystem is an orthonormalbasis

We have an orthonormalsystem in $L^2(0, 2\pi)$: $\{e^{ikx} : k \in \mathbb{Z}\}$. Now I want to show that it's also an orthonormalbasis. I thought the easiest way to do that would be to show that ...
3
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1answer
27 views

Weak compactness of a set of translates in $C_0(\mathbb{R})$

Let $f \in C_0(\mathbb{R})$. Consider the set of translates of $f$ $$ A = \{ f_t : t \in \mathbb{R} \}$$ where $f_t(x)=f(x+t), x\in \mathbb{R}$. I want to show that $A$ is relatively compact in the ...
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1answer
20 views

Is adjoint space different from dual space? [on hold]

For a linear space X , whether the dual space and adjoint space are the same or different?
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32 views

Direct Integral: Dimension

Direct Integral Given a Borel space $\Omega$ with measure $\mu$. Given Hilbert spaces $\mathcal{h}_x$ for $x\in\Omega$; set $\mathcal{h}:=\bigcup_{x\in\Omega}\mathcal{h}_x$. Regard the function ...
2
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1answer
33 views

Fundamental solution for a parabolic PDE with costant coefficents

as it is well known, the fundamental solution of the heat equation is the function $G(t,x)=\frac{1}{(4\pi t)^{n/2}}e^{\frac{|x|^2}{4t}}$, for all $t>0,x\in\mathbb{R}^n$. I wonder if exists (and ...
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0answers
39 views

Physical meaning of Rudin's equation in Hilbert space

Rudin's Functional Analysis, p. 334, Corollary of Theorem 13.10 says Corollary If $a\in H$ and $b\in H$, the system of equations $$-Tx+y=a$$ $$x+T^*y=b$$ has a unique solution with $x\in ...
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0answers
27 views

Prob. 10, Sec. 4.5 in Kreyszig's Functional Analysis: How to relate this result to solution of equations?

Let $T \colon X \to Y$ be a bounded linear operator, where $X$ and $Y$ are normed spaces, both real or both complex; let $B$ be a subset of the dual space $X^\prime$ (i.e. the normed space of all the ...
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0answers
26 views

Continuity of translation property [duplicate]

Let $u \in L^{p}(U)$ where $1 \leq p \lt \infty$ & $U \subseteq \mathbb R^{n}$ . Define : $F : \mathbb R^{n} \to L^{p}(U) $ by $ F(y) := u(x+y)$ . Prove that: as a function of $y$ ; $F(y) $ is ...
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0answers
28 views

Consider $c_{00}$ as a subspace of $(\ell^p,\|\cdot\|_p)$. Show that the closure of $(c_{00},\|\cdot\|_1)$ is $\ell^1$

Consider $c_{00}$ as a subspace of $(\ell^p,\|\cdot\|_p)$. Show that the closure of $(c_{00},\|\cdot\|_1)$ is $\ell^1$, closure of $(c_{00},\|\cdot\|_2)$ is $\ell^2$ and closure of ...
3
votes
1answer
35 views

Question on Lp spaces defining metric

First question here so really excited and hope you can help me, thanks! In my intro to functional analysis class we just now covered $L^p$ spaces and I was presented with this homework question: ...
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4answers
440 views

Why isn't every element of the spectrum an eigenvalue? (Where is the error in my proof?)

My book defines the spectrum like this: Let $H$ be a complex Hilbert space, let $I \in B(H)$ be the identity operator and let $T \in B(H)$. The spectrum of $T$, denoted $\sigma(T)$, is defined ...
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0answers
26 views

Statement about the discrete (metric) space, and both an open and closed ball.

I have the following statement from my notes: "Let $(X,d)$ be the discrete space i.e. any non-empty set with the discrete metric ($d_d(x,y)=1$ for all $x\neq y$). Then, amazingly, $B_1(x)=\{x\}$, a ...
0
votes
1answer
51 views

$Hom(V,W)$ remains unchanged when norms of $V$ and $W$ are replaced with equivalent norms.

I was thinking about the following question from section 3.4 of Loomis and Sternberg's Advanced Calculus The fact that $Hom(V,W)$ is unchanged when norms are replaced by equivalent norms can be ...
3
votes
2answers
44 views

Prove that the linear transformations are the same.

I have this lemma: If X is a complex inner product space and $S,T \in B(X)$ are such that $(Sz,z)=(Tz,z)\forall z \in X$, then $S=T$. $B(x)$ is the set of bounded linear operators from X to X. ...
3
votes
1answer
31 views

Show that there exists a sequence of functions $\{f_n:[0,1]\to\mathbb C\}$ satisfying the given condition.

Show that there exists a sequence of functions $\{f_n:[0,1]\to\mathbb C\}$ satisfying: 1) $f_n\to0$ pointwise; 2) $\gamma_nf_n\not\to0$, for all $\gamma_n\in\mathbb C$ such that ...
2
votes
1answer
29 views

One norm is stronger than but not equivalent to another norm

Let $X=C^{1}([0,1])$. For $x \in X$, let $$||x||_{0}^{'}=|x(0)|+||x'||_{\infty}$$ I need to show that $||.||_{0}^{'}$ is stronger than but not equivalent $||.||_{\infty}$. It is easy to see that ...
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1answer
27 views

Prob. 3, Sec. 3.2 in Kreyszig's Functional Analysis Book: Is the space of all polynomials of a fixed degree complete? [duplicate]

Let $n$ be a given natural number, and let $X$ denote the vector space consisting of the zero polynomial and of all polynomials of degree at most $n$, with real or complex numbers as co-efficients, ...
0
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0answers
17 views

Showing the space of $\beta$-operators between Banach lattices is a Banach space [on hold]

Let $E,F$ be a Banach lattices. A linear map $T:E \rightarrow F$ is called $\beta$-operator if $\lVert T\rVert_\beta < \infty$, where $$\lVert T\rVert_\beta := \sup \bigl\{ \bigl\Vert ...
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0answers
30 views

Is this function uniformly convex?

I am using the following definition of uniformly convex: A continuous functional $G:Y \to R$ is uniformly convex on a ball $$B(0,\delta):=\{y\in Y : \|y\| < \delta\}$$ if there exists a ...
1
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1answer
32 views

Is it true $\lambda_i(f(A))=f(\lambda_i(A))$ for non-negative and non-decreasing function on $[0,\infty)$

I am wondering is it next true: Suppose that $f(t)$ is non-negative and non-decreasing function on $[0,\infty)$ and let $A$ be a positive operator on some infinite-dimensional separable Hilbert ...
0
votes
0answers
35 views

The space of continuous functions as a dual space

Let $X$ be some topological Hausdorff space and $C_b(X)$ the space of bounded complex continuous functions on $X$. Is there a Banach space $B$ such that $B^* \simeq C_b (X)$? I know of a very similar ...
3
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1answer
32 views

Convergence that preserves smoothness

One of the advantages of uniform convergence is that it preserves continuity (among other properties). Unfortunately, it does not preserve derivability. Is there a convergence mode preserving it?
3
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1answer
21 views

Weakly compact operator with different domains

Let $A$ be a Banach algebra. Suppose that $e\in A$ such that $e^2=e$ and $eAe$ is division algebra(i.e., $eAe$ is unital and every element of $eAe$ has inverse in $eAe$). Define $T_e:A\to A$ with ...
3
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1answer
19 views

Question about assumptions for Picard-Lindelof Theorem in Zeidler's functional analysis text

In Zeidler's text on functional analysis pg.24 he wrote... The Picard Lindelof Theorem: Assume the following: (a) the function $F: S \to \mathbb{R}$ is continuous and the partial derivative ...
2
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1answer
34 views

Help understanding Rudin's proof showing that $C_c(X)$ is dense in $L^p(\mu)$

The proof is from Rudin's "Real and Complex Analysis." It states For $1\leq p<\infty$, $C_c(X)$ is dense in $L^p(\mu)$ The proof is Let $S$ be the class of all complex, measurable, simple ...
2
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0answers
18 views

Why does minimizing $H[f] =\sum^{N}_{i=1}(y_i-f(x_i))^2+\lambda \| Pf \|^2 $ leads to solution of the form $ f(x) =\sum^N_{i=1}c_iG(x; x_i)+p(x)$?

I was reading the following paper of dimensionality reduction (1) and also one on theory of networks for approximations and learning (2) and was trying to understand how the regularization problem ...
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0answers
15 views

finite differences nonlinear least squares

I am facing to following nonlinear least-squares problem: $$\min_{u,\gamma} \frac{1}{1000} \int_{ \gamma(x,y)^2} + \int_{ [u(x,y)−u (x,y)]^2} + \int_{ [∆u(x,y)−\gamma(x,y)u(x,y)]^2}$$ where the ...
3
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1answer
56 views

Prove that ${\{f_n\}}_{n\in\Bbb{N}}$ has a subsequence that converges uniformly to a continuous function on $[0,1]$

Consider the sequence ${\{f_n\}}_{n\in\Bbb{N}}$, where for each $n\in \Bbb{N}$ the function $f_n:[0,1] \to \Bbb{R}$ is absolutely continuous and satisfies $f_n(0)=13$ and $$\int_{[0,1]}|f_n'|^4dx ...
1
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1answer
82 views

Equivalence of holomorphic functions

Given that $$\left(1-\frac{z}{\zeta_j}\right)^{-z}=\sum\limits_{k=1}^\chi\frac{z^k}{k\zeta_j^k},$$ where $\chi$ is the largest nonnegative integer $k$ for which ...
2
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1answer
53 views

The sequence $f_n=x^n$ is not weakly convergent in $C[0,1]$

Let's consider the sequence $f_n=x^n$ for $n \in \mathbb{N}$ in $C[0,1]$ equipped with the usual supremum norm. How can we show that $f_n$ does not converge weakly in $C[0,1]$ without using an ...
0
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0answers
52 views

Weak convergence and convergence almost everywhere

If a bounded sequence $(u_n)$ converge weakly to $u$ in $W^{1,p}(\Omega)$ (Where $\Omega$ is an open bounded from $\mathbb{R}^N$ and $N>p$) Have we that $u_n(x)$ converge to $u(x)$ almost ...
3
votes
1answer
25 views

What is the $w^{*}$-closure of the finite rank operators in $B(H)$?

I know that the norm closure of the finite rank operators on a Hilbert space is the compact operators $K(H)$. I've been trying to determine what is the $w^{*}$-closure but I am not getting any good ...
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0answers
72 views

Alternative ways to prove $\{f:f(0)=\sum_k f(\frac{k}{\sqrt{n}})g_n (k)\}$ is dense in $\{f\in C^2 (\mathbb{R}) : f(0)=\int_{\mathbb{R}} f(u)g(u)du\}$

I want to prove that $$E:=\bigcap_{n\geq 1} \left\{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f\left(\frac{k}{\sqrt{n}}\right)g_n (k)\right\}$$ is a dense subset of: $$F:=\left\{f\in C^2 (\mathbb{R}) ...