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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25 views

When are the marginals of an extremal invariant measure also extremal invariant?

Let's suppose that $X$ is a compact metric space, and thus as is $X \times X$. If given a Markov process on $X \times X$ with marginals that are Markov processes on $X$, then we know that the ...
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45 views

Is the $C^r(M, N)$ space, with the strong (Whitney) topology, a Fréchet-Urysohn space?

Given smooth, non-compact manifolds $M$ and $N$, consider the function space $C^r(M, N)$. Equipped with the strong (Whitney) topology, this space is Hausdorff and Baire. It is, however, not first ...
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40 views

Fourier transform of $e^{it\sqrt{a^2+x^2}}$

The question is clear, I came up with this Fourier transform to calculate while searching explicit solutions for a PDE, but I don't even know if it is feasible. $ \mathcal{F}_x(e^{it\sqrt{a^2+x^2}}) ...
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28 views

Show: the subspace of compact operators is $\Lambda$-invariant

Let $X$ be a Hilbert space and $A \in \mathcal{L}(X)$ be fixed. Define $\Lambda: \mathcal{L}(X) \rightarrow \mathcal{L}(X)$ by $ \Lambda (T)=A^{*}T+TA,\;T\in \mathcal{L}(X)$. Show: the subspace of ...
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68 views

the sum of closed convex sets

Let C and D be two closed convex subsets of a Banach space with C+D is closed. If bounded sequence $\{x_n\}\subset C+D$, can we choose bounded sequences $\{c_n\}\subset C$ and $\{d_n\}\subset D$ such ...
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29 views

Limit of exp of self-adjoint operator

Let $A$ be self-adjoint (possibly unbounded) operator on Hilbert space $\mathcal{H}$. Under what conditions $w-\lim_{t\rightarrow\infty} e^{i A t}=P_0$, where $w-\lim$ - the limit in weak operator ...
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39 views

Show that the map $T(x)=x/2+x^{-1}$ is a contraction and find $\alpha$

Let $X=[1,\infty)$. Show that the map $T(x)=x/2+x^{-1}$ is a contraction, and find $\alpha$. Proof: A function $T:X \rightarrow X$ is said to be a contraction if $dist(T(x),T(y)) \leq \alpha ...
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57 views

Open Convex Subsets of Dense Spaces

So I asked this question yesterday, Existence of Non-Trivial, Convex, Open Set in $C_{\mathbb{C}}[0,1]$ Under $L^{0}$ Metric, and it made my start wondering the following... Suppose the following: ...
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70 views

Riesz Representation Theorem

I am unfamiliar with Quantum Mechanics and all that stuff. I have recently studied Riesz Representation Theorem , I got to know that it justifies ket and the bra notation. Can anyone please give an ...
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36 views

Are the continuous linear functions from a norm space to R bounded?

$\{X, \|\cdot\|\}$ a normed space, a function, maping from $X$ to $\mathbb R$, is linear and continuous. Is it a bounded linear function?
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35 views

Question about Schauder basis

The question is : Let $B$ be a Banach space and suppose $\{x_n\}$ the Schauder basis and $M$ be the space of sequence of scalars $\{a_n\}$ such that the sup norm of power series of $a_n x_n$ ...
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37 views

Proof that these are Fourier coefficients

I proved that for $f \in \ell^1 (\mathbb Z)$ its Gelfand transform $\widehat{f}$ is a map $\widehat{f}: S^1 \to S^1$ defined by $$ \widehat{f}(z) = \sum_{n \in \mathbb Z}f(n) z^n$$ In Murphy's book ...
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44 views

Question about Hahn-Banach separation Theorem

So here is my question, I am just reading about the Hahn-Banach separtion Theorem and there is one case where a question appeared, namely, Let $X$ be a normed $\mathbb R$ vectorspace and let $A,B$ ...
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46 views

Examples of semigroups of contractive Fourier multipliers but not positive?

Can you show me a concrete an example of semigroup $(T_t)_{t\geq 0}$ of Fourier multipliers such that each operator $T_t$ induces a contractive Fourier multiplier $T_t\colon L^p(\mathbb{T}) \to ...
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62 views

Equivalent to not separable?

I wanted to ask if this statement is true, Let $X$ be a normed vectorspace. Then, X is separable iff every disjoint, and open familiy of subsets is countable. My idea to prove it was the ...
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55 views

Transfer vector space properties to dual space

I am curious about this here (Actually, I don't know if my assumptions are true or not) a) Let $X$ be a Banach space that is isomorphic to $Y$, then $X^*$ is also isomorphic to $Y^*$. I sketched a ...
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41 views

If the quotient of a subspace of a banach space is finite, is it a closed subspace?

Given a Banach space B,V is a subspace of B,if B/V is finite dimension,then is it enough to show that B is closed? Thanks!
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30 views

Showing a particular integral operator is trace class

Let $f$ and $P$ be continuous, integrable functions $\mathbb{R} \to \mathbb{C}$ vanishing at $\pm \infty%$. Concisely, $f,P \in C_0(\mathbb{R}) \cap L^1(\mathbb{R})$. Also, assume that $P$ is ...
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78 views

the dual space of $C([0,1])$

I'm studying Conway's Functional Analysis by myself. the following question is one of this book's Exercise. If $n\geq 1$ , does there exist a measure $\mu$ on $[0,1]$ such that $\int pd\mu =p(0)$ for ...
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76 views

Fractional Sobolev spaces and weighted L2 spaces

For $s\in[0,1]$ define function spaces $H^s(\mathbb{R})=\{u\in L_2(\mathbb{R}): (1+|\cdot|^2)^{s/2}\mathcal{F}u\in L_2(\mathbb{R}) \}$ (where $\mathcal{F}$ denotes the Fourier transform) i.e. the ...
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52 views

What is the product of bessel functions of first and second kind when their arguments are same and tends to zero?

As we know, $\lim_{x \to 0} J_m(x)=0$ where $m\geq 1$ and $\lim_{x \to 0} Y_m(x)=\infty$ then what would be $\lim_{x \to 0}J_m(x)Y_m(x)$. Matlab shows the product is finite and $< 1$. What should I ...
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32 views

Proving the Rietz-Fischer Theorem for $p = \infty$

Rietz-Fischer Theorem: Let $E$ be a measurable set and $1 \le p \le \infty$. Then every rapidly Cauchy sequence in $L^p(E)$ converges both with respect to the $p$-norm and pointwise almost everyone ...
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27 views

Conjugating an operator with a gauge transformation; how is the kernel affected.

For the differential operator $$ D := i I \frac{d}{dx} + A(x) \colon C^\infty_T([0,\beta]),\mathbb{C}^m) \to C^\infty_T ([0,\beta],\mathbb{C}^m) $$ where $A(x)$ is Hermitian and $C^\infty_T ...
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60 views

Don't understand a PDE argument involving $L^p$ norms and inequalities

I'm reading this paper. I do not understand how the author proves Corollary 5.12 for the case $p=2$. He addresses everything except $p=2$ but claims it holds for all $p$. Can someone help me to see ...
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38 views

The uniqueness of a solution of a system of differential equations

Suppose I have a system of delay differential equations $\frac{d G_1(\phi_1(s))}{ds}= F_1(\phi_1(s),\phi_2(1-s),1-s)$ $\frac{d G_2(\phi_2(1-s))}{ds}= F_2(\phi_1(s),\phi_2(1-s),s)$ where $G_1, G_2, ...
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11 views

invariant measures for the independently coupled Feller process

If I take countably many Markov processes and couple them independently, how does the collection of invariant measures for the coupled process relate to those of the marginals? I conjecture that it ...
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75 views

Expectation of $p$-norm under a Gaussian on the Hilbert space $L^2(S^1)$

Let $\mu$ be a centered Gaussian measure with (nondegenerate) covariance $Q$ on the Hilbert space $L^2(S^1;\mathbb R)$ where $S^1$ is the circle. We can take for example the covariance ...
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51 views

Analytic solutions of an equation

I am trying to find analytic solutions of this equation for $x$ with parameter $a$ ($x>0, a>0$): ...
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26 views

Schauder's theorem: consequences and applications

I am about to give an informal talk about Schauder's theorem ($T:X\to Y$ linear operator between Banach spaces is compact if and only if its adjoint is). Does anyone know any derived ...
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49 views

The infinitessimal generator of Brownian Motion

Background: I know, and can almost prove, that the infinitessimal generator of BM is the Laplacian/2. For me (and I have never heard it done differently) we have Feller Processes, of which BM is one ...
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43 views

When is a delta function a valid distribution?

If $f:\mathbb{R}^k \rightarrow \mathbb{R}$ is nicely behaved, one can view $\delta(f)$ as a distribution (linear functional on $C^{\infty}_c(\mathbb{R}^k)$)- but what if you don't have nicely behaved ...
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50 views

Need help understanding this proof

I need help understanding the following: If $A$ is a (complex) banach algebra and $I$ is a proper modular ideal then $\overline{I}$ is also proper. Proof. Let $u\in A$ be such that $a-ua, a-au \in ...
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39 views

Supermodularity and n-increasingness

Let $\geq$ be the usual partial order over $\mathbb{R}^n$ (i.e. if $x,y\in\mathbb{R}^n$, $x\geq y$ iff $ x_i\geq y_i \forall i=1,\dots,n$). Definition 1: Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$. ...
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37 views

Contraction Mapping Conditions

Consider $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ continuous and uniformly bounded, and consider the iteration $$ x^{(k+1)} = \frac{1}{k} \sum_{i=1}^{k} f\left( x^{(i)} \right) + x^{(i)} $$ for ...
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38 views

Want to determin a dual space

I would like to determine the dualspace of some normed vectorspace. Namley, $$c_0:=\{x=(x_n)_{n\mathbb N}\subset\mathbb R:\lim_{n\rightarrow\infty}x_n=0\; \text{ and } ||x||=\sup_n|x_n|\}$$ I ...
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50 views

What are these spectra (part 1)

I needed some insight into spectra in Banach algebras. To this end I tried to calculate a few examples. I will post them not all in one post. Could somebody please tell me if they are correct? Here ...
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43 views

a complete space

Define a set $$X=\left\{f:\mathbb{R}\rightarrow\mathbb{R}|f \mbox{ is n-times continuously differentiable}\right\}$$ equipped with the norm $$||y||=\max_{\begin{subarray}{l} ...
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64 views

existence and unicity of solution of an Variational Formulation

Let $\Omega=\mathbb{R}_+$ and let $\Gamma=\{(x,0), x \in \mathbb{R}\}$ We consider the problem $$ \begin{cases} & - \mathrm{div} (A \nabla u) + \lambda u = f, \quad (x,y) \in \Omega\\ & ...
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116 views

Multidimensional (Fredholm) integral equation (of first kind)

Suppose, \begin{align*} g(t_1,t_2)=\int f(s) \left[K_1(t_1,t_2,s)h(t_1) + K_2(t_1,t_2,s)\right]ds %\\ %g(t_1,t_2)=\int f(s) \left[K_1(t_1,t_2,s) + K_2(t_1,s)h(t_2)\right]ds \end{align*} The problem is ...
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40 views

Differentiabilty of certain convolution

Let $\Gamma$ be a smooth closed curve. Suppose $f\in L^2(\Gamma,ds)$ and $g$ is defined everywhere on $\mathbb{C}$ with compact support. Moreover $\frac{dg}{dz}$ exists everywhere but point $z=0$. ...
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81 views

Help with Gronwall's Inequality

I'm trying to solve an extra credit hw problem that has to do with Gronwall's Inequality. I understand (or think I do) how to find an upper bound when $p=0$, but I'm unsure of how to handle the extra ...
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48 views

Uniformly Bounded in a Schauder Norm

Suppose $\{h_{k}\}$ is a sequence of elements in $C^{2,\alpha}$. I want to show that $\{h_{k}\}$ is uniformly bounded in $C^{2,\alpha}$. Now, if I'm not mistaken, then $$||h_{k}||_{\displaystyle ...
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110 views

Proving that a positive operator has a unique square root

Rudin's functional analysis page 331 theorem 12.33) proves this. He proves uniqueness by 'going back' to the general algebra setting. I was just wondering whether there is a more direct way of doing ...
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43 views

Lack of a polar decomposition

Prove that the left and right shifts on $l_{2}$ have no polar decomposition (i.e. $UP$ where $U$ is unitary and $P$ is positive).
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18 views

Normed and 2-normed spaces

Is every normed space a 2-normed space? Again, is every 2-normed space a normed space? please explain. Thanks in advance!
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39 views

Characterizing elements in $X^{\ast}\hookrightarrow L^{\infty}(G)$

I am in the following situation: Let $G$ be a locally compact group (possibly Hausdorff). Let $\Phi:L^{1}(G)\twoheadrightarrow X$, where $X$ is a Banach space. Then ...
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138 views

Find features in a Signed Distance Field

I'm currently trying to improve a meshing algorithm for signed distance fields (which are of the form $f(x,y,z) = w$, where $x,y,z$ is the location of my query, $w$ indicates the distance to the ...
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40 views

Self-Adjointness of beam stiffness operator

I think it is well known that the operator $\frac{EI}{\rho} \frac{\partial^4}{\partial x^4}$ which arises from a standard Euler-Bernoulli beam is self-adjoint in $H$, where $H = L^{2}$, given ...
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51 views

Analytic family of operators?

If $E_{\lambda}, F_{\lambda}$ are two families of complex Hilbert spaces and $L_{\lambda} : E_{\lambda} \rightarrow F_{\lambda}$ is a family of bounded linear operators, where $\lambda$ is a complex ...
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32 views

The Haar basis ,proof of orthonoramality.

please i have this problem and i known how to prove completeness but do not know how to prove that it is orthonormal. I will appreciate it if anyone can help me. Given that $n\geq1$ write ...