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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Differentiating an infinite series in Hilbert space

Suppose $H$ is separable Hilbert space and $w_j$ is a basis. Suppose we have $h=\sum a_j(t)w_j$ an infinite sum where the coefficients are functions of $t$. The sum makes sense in the sense that the ...
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89 views

Concerning unbounded linear operators on a Hilbert space

Let $H$ be some Hilbert space and let $B:H\rightarrow H$ be a bounded linear operator and $T:H\rightarrow H$ an unbounded linear operator. Furthermore we assume that $T$ is closed ,i.e. it's graph in ...
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80 views

$f(a+b,\lambda) = f (a,\lambda) \cdot f(b,\lambda)$

Consider the equation: $f(a+b,\lambda) = f (a,\lambda) \cdot f(b,\lambda)$, for $a \geq 0$ and $b \geq 0$. Is my understanding that this simple functional equation is important in analysis. Can ...
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30 views

what to do if it's not direct sum?

Suppose $X=Y+Z$ is Banach, $Y$ and $Z$ are closed subspaces. I want to show there exists $\alpha>0$ such that $\forall x \in X, \exists$ $y \in Y$ and $z \in Z$ such that $x=y+z$ and $\|y\|+\|z\| ...
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53 views

A space $X$ that contains a copy of $\ell_1$, does not contain a complemented copy of $\ell_1$, and whose dual is not weakly sequentially complete

I want to find an example of a Banach space $X$ which contains a copy of $\ell_1$, does not contain a complemented copy of $\ell_1$, and so that $X^*$ is not weakly sequentially complete.
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231 views

Topics of advanced functional analysis

I did a course in introductory functional analysis and liked it. Now I want to learn more functional analysis with the goal of maybe eventually doing research (phd). I tried to find out what ...
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151 views

How to decompose a representation into direct sum of cyclic representation?

Let $U$ be the bilateral shift operator on $l^2 (\mathbb Z)$, let $T=U+U^*$. How to calculate $\sigma(T)$? And how to show there is no cyclic vector for the action of $C^*(T,I)$. Further how to ...
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184 views

Inverse of Identity plus Volterra operator

consider the following operator or $L_2(0,1)$, $(Pw)(x)=w(x)+\int_0^x K(x,y)w(y)dy+\int_x^1 K(y,x)w(y)dy$, where the integral kernel is a polynomial. I am trying to construct the inverse of this ...
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71 views

Find Lipshitz constant

Let $x, y$ be two vectors in $R^n$. Let $\pi$ be permutation on $\{1, \ldots, n\}$ with uniform distribution. Find Lipshitz constaants of the following functions: $$ f(\pi)=\sum_{i=1}^nx_{\pi(i)}y_i ...
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69 views

Different functional brachystochrone

Until today I thought that $$ \int_0^b \sqrt{\frac{1+y'(x)^2}{2gy(x)}} dx$$ would be the only functional to derive the brachystochrone, but in the textbook Variational Methods in Mathematical Physics ...
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55 views

Fredholm alternative for polynomially compact operators

Let $T \in \mathcal{L}(E)$ be a polynomially compact operator, i.e., there exists a polynomial $p$ such that $p(T)$ is a compact operator. Suppose $p(1) \neq 0$. I want to show that $N(I-T) = \{0\} ...
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88 views

Open map in Banach algebra

I'm having trouble showing a certian function is open and can be extended. Let $\Omega$ be a completely regular topological space and $A=C_b(\Omega)$ the space of all complex-values bounded ...
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117 views

Biduals generated by projections

This question is motivated by a similar question recently posed at MO: http://mathoverflow.net/questions/122091/masas-in-second-duals-of-banach-algebras In this setting, let $B$ be a Banach algebra ...
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58 views

Some properties of an analogue of the integral Fourier operator

Let $\phi(x,\theta)$ be an infinitely differentiable function in $X^2 = X \times X$, where $X = \mathop{\mathsf{int}}\mathbb{R}^n_{+}$, and let$\phi(\lambda x, \theta) = \phi(x,\lambda \theta) = ...
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401 views

The spectrum of the shift operator

I come across this problem in Conway's functional analysis: The forward shift $S:l^2\rightarrow l^2$ takes $e_n$ to $e_{n+1}$ for all $n\in N$.In this case $S^*$ is the backward shift. So what is the ...
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64 views

Banach dual space integral

Let $X$ be a Banach space and $f_t \in X^*$ for each $t \in [0,t_0]$. Suppose that $$\int_0^{t_0} f_t(x) = 0$$ for all $x \in X$. 1) Does it make sense to write $\int_0^{t_0}f_t = 0$? 2) If so, does ...
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72 views

Invariant spaces for a shift semigroup

Consider the space $C_0 := \{f \in C[0,1], f(0)=0\}$ and the shift semigroup $(T^t, t \in [0,1])$, defined by $T^t f(x) := \begin{cases} f(x-t), & t\le x\\ 0, & \text{otherwise} \end{cases}$ ...
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81 views

Invertibility of a Toeplitz operator

Let $\phi$ be a real-valued function. I am trying to show that the Toeplitz operator $T_\phi$ is invertible if the function 1 is in the range of $T_\phi$. Here is what I got so far: There exists a ...
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79 views

The control of norm in quotient algebra

Let $B_1,B_2$ be two Banach spaces and $L(B_i,B_j),K(B_i,B_j)(i,j=1,2)$ spaces of bounded and compact linear operator between them respectively. If $T \in L(B_1,B_1)$, we have a $S \in K(B_1,B_2)$ and ...
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141 views

What is the dual space of $C([0,T];X)$ ($X$ Hilbert space)?

What is the dual space of $C([0,T];X)$, where $X$ is a Hilbert space? Is it $\operatorname{BV}([0,T]; X^*)$? As we know, for $C([0,T])$, the dual space is $\operatorname{BV}([0,T])$, but when it is ...
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173 views

The converse of James's Theorem

The famous James theorem states that: Theorem. Let $X$ be a (Hausdorff separated) locally convex space (LCS for short) with topological dual $X^*$ and let $B\subset X$ be weakly-closed. If $X$ is ...
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356 views

Confused by a proof in Rudin *Functional Analysis*

I am reading Rudin's Functional Analysis and got quite confused by his proof of Thm 8.5, that is, the existence of fundamental solutions for differential operator $P(D)$, where $P$ is a polynomial. ...
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128 views

Haar Measure: Unimodular Locally Compact Groups

I have the following problem: "Let $G$ be a locally compact group, all of whose normal subgroups are contained in $Z(G)$. Prove that $G$ is unimodular." My attempt at attacking the problem was to ...
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251 views

Can infinitesimal generator be defined by the time-inhomogeneous stochastic process?

The following is the definition of infinitesimal generator from Oksendal. Let $\{X_t,t\in[0,T]\}$ be a time-homogeneous It\^o diffusion in $\mathbb{R}^d$. The $\textit{infinitesimal generator}$ ...
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123 views

Prove that a flat shape minimizes a functional

The following functional arises in an information theoretic problem that I work on currently. $$I(G(\omega)) = \int_{-\kappa\pi}^{\kappa\pi} \frac{A}{G(\omega)+A}d\omega-\frac{\left| ...
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107 views

A question regarding vector spaces with partial order

$\newcommand{\N}{\mathbf{N}}$$\newcommand{\R}{\mathbf{R}}$ Let $X=(X,\leq)$ be a Riesz space (a lattice which is also an ordered vector space over $\R$). Define $X^+=\{x\in X\colon x\ge 0\}$. Are the ...
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283 views

Continuous spectral value of right shift operator $\ell^2(\mathbb{N})$

Let $T:\ell^2(\mathbb{N} \to \ell^2(\mathbb{N})$ be the operator that sends$(x_1,x_2,x_3,...) \to (0,x_1,x_2,x_3,....)$. I want to show that $\lambda = 1$ is in the continuous spectrum. To approach ...
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131 views

Distributional derivative of bounded functions

Let $f$ be a bounded measurable function on $\mathbb{R}$. We may consider its derivative $f'$ as a distribution on $\mathbb{R}$. Is there a reasonable description of those distributions $\psi$ ...
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122 views

Does such an operator exist?

Suppose $T$ is a bounded operator on $H:=\mathcal{l}_2$ which is quasinilpotent and has the property that both $T$ and $T^{*}$ are not injective and have finite dimensional kernels. Is it possible ...
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312 views

Eigenprojection as Contour Integral over Resolvent

Let $H$ be a Hilbert space and let $A \in L(H)$ be a bounded linear operator. Assume that $\lambda$ is an eigenvalue of $A$ and assume further that $C_\lambda$ is a simple closed curve in the complex ...
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109 views

Relations between spectrum and quadratic forms in the unbounded case

Let $H$ be a complex Hilbert space. If $B$ is a bounded self-adjoint operator on $H$ then its spectrum is a closed and bounded subset of the real line and we can find its extremes in terms of the ...
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104 views

Direct limits of completely positive maps on $C^*$-algebras vs. operator systems

I believe I've heard, as part of the "lore," that the category (operator systems, completely positive maps) has direct limits, whereas the category ($C^*$-algebras, completely positive maps) does not. ...
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114 views

Convergence of integrals of Radon measures

Let $X$ be a locally compact Hausdorff space and let $\mu_n$ be a sequence of bounded variation Radon measures on $X$ such that $\int_X g \;d\mu_n \rightarrow \int_X g \;d\mu$ for each $g \in C_0 (X)$ ...
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231 views

Reference for Martingale version of Riesz representation theorem / Riemann–Stieltjes integral

(Please let me know if this is more appropriate as a MathOverflow question.) I can work out most of the following martingale generalization to the Riesz representation theorem and the ...
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937 views

Integral of the derivative of a function of bounded variation

Let $f\colon [a,b] \to \mathbb R$ be of bounded variation. Must it be the case that $|\int_a ^b f' (x) |\leq |TV(f)|$, where $TV(f)$ is the total variation of $f$ over $[a,b]$? If so, how can one ...
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votes
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231 views

Weakest topology with respect to which ALL linear functionals are continuous

One often considers a Banach space $X$ under the "weak topology", ie. the weakest topology such that all bounded linear functionals are continuous. This leads me to wonder about the weakest topology ...
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161 views

Fractional iteration of the Newton-approximation-formula: how to resolve one unknown parameter?

For my own exercising I tried to find a closed form expression for the Newton-approximation algorithm, beginning with the simple example for getting the squareroot of some given $ \small z^2 $ by ...
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177 views

Vector valued contraction

I am familiar with the Banach Fixed Point Theorem and I have used it to prove existence and uniqueness of functional equations in Banach spaces like $C(X)$, the space of bounded continuous function ...
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0answers
41 views

A possible Corollary of the Fredholm alternative?

Let $H$ be a Hilbert space, $P : H \rightarrow H$ a positive-definite (bounded) operator and $K : H \rightarrow H$ a compact (not necessarily self-adjoint) operator. Let $T = P + K$. In particular, ...
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votes
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27 views

give necessary and sufficient conditions that every functional in $w^*-cl M$ be the $w^*$- limit of a sequence from M

Let $X$ be a separable Banach space. If $M$ is a linear manifold in $X^*$ give necessary and sufficient conditions that every functional in $w^*-\mathrm{cl} M$ be the $w^*$- limit of a sequence from ...
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55 views

Using Stone–Weierstrass theorem for completely regular space

Let $X$ be completely regular. If $K$ is a compact subset of $X$, define $$p_K(f)=\sup\{|f(x)|;x\in X\}$$ then $\{p_k; \text{K is a compact }\}$ is a family of seminorms that makes $C(X)$ into a ...
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123 views

A limsup version of the Principle of Uniform Boundedness?

Suppose $X$ is a Banach space and let $\{f_\alpha\}$ be a net of continuous linear functionals satisfying $\limsup_{\alpha} | f_\alpha(x) | < \infty$ for each fixed $x \in X$. Is it true that ...
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77 views

Positive Operators: Definition?

Let $A$ be a self adjoint element of a C*-algebra $\mathcal{A}$ resp. a self adjoint operator of the operator algebra $\mathcal{B}(\mathcal{H})$ of bounded operators over a Hilbert space ...
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50 views

Is $L^p$ separable?

Whether a $L^p(X,\mu)$ space is separable? I understand that the answer depends on $p$ and $X$. It seems to me that it is separable when $1\leq p < \infty, X=\mathbb{R}^n$ or $X=\mathbb{N}$. ...
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81 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 ...
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25 views

On maximal ideal spaces of a banach algebra

I am reading this article on maximal ideal spaces and there is this part that I don't quite understand very well, hope you guys can help me out. "Let $M(A)$ denote the maximal ideal space of a ...
3
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107 views

Asymptotically isometric copy of $\ell_1$

A Banach space $X$ is said to contain Asymptotically isometric copy of $\ell_1$ if there is a null sequence$(\rho_n)_{n=1}^{\infty}$ in $(0,1)$ and a sequence$(x_n)_{n=1}^{\infty}$ in $X$ so that ...
3
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39 views

Minimum regularity Of Stoke's theorem to hold in smooth manifold.

Stokes’ Theorem on Manifolds is often express as follows: Given a differential m-form $\omega$ whose support is the $C^{\infty}$ $m$-dimensional compact manifold ${\cal{M}}$ with boundary ...
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41 views

Vector Space Verification

I just took an exam asking me if the following are a vector space over $\mathbb{R}$ assuming that the set of all real valued functions on the interval $[0,1]$ is a vector space with theoperations ...
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44 views

Do we have $C^\infty \cap \mathcal{O}_C' = \mathcal{S}$ and/or $C^\infty \cap \mathcal{S}' = \mathcal{O}_M$?

We define the following traditional function spaces from distribution theory. $\mathcal{S}$ the space of rapidly decreasing smooth functions. $\mathcal{S}'$ the space of tempered distributions, dual ...