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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Weak topology on $L^p,~p> 1$

How looks like the weak topology in the particular case $X=L^p$, I mean, is possible to detail this topology beyond standar form: Arbitrary union of finite intersections open pre-images of opens ...
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51 views

Strongly convergence in L^2

Let the sequence $(f_n(x,u))$ such as, for all $n,$ $f_n$ is Caratheodory, and $|f_n|\leq g$ where $g \in L^1(\Omega)$ Let $u_n \in H^1_0$ such as it is strongly convergent to $u$ in $L^2$ and a.e ...
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127 views

Rising Sun Inequality (Dunford-Schwartz maximal inequality)

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be an absolutely integrable function, and let $f^*:\mathbb{R} \rightarrow \mathbb{R}$ be the one-sided signed Hardy-Littlewood maximal function $$f^*(x) := ...
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108 views

How to get a grip on codimensions

I am trying to find a proof for the following problem: Let $X,Y$ be Banach spaces $A,B:X \rightarrow Y$ are bounded linear operators $Ran(A)$ is closed, and $\dim(\mathrm{Ker}(A))$ or ...
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285 views

Properties of the square norm in Banach spaces

Let $X$ be a Banach space with its dual $X^*$. Consider the mapping $f: X\rightarrow \mathbb{R}$ given by $$ f(x)=\frac{1}{2}\|x\|^2. $$ We have know that when $X$ is a real Hilbert space ($X=X^*$) ...
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147 views

Is this Neumann series solution unique?

I have a Fredholm integral equation of the second kind given as $$f(x)=g(x)+\lambda\int_{-\infty}^\infty K(x,y)f(y)dy, $$ where $\lambda\in(0,1)$, the kernel $K(x,y)=\phi(x-y)$ is a Gaussian ...
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64 views

Equivalence of definitions of $C^k(\overline U)$

let $U$ be an open set of $\mathbb{R}^n$, that contains at least some open set. In Evans book we find the definition $$C^k(\overline U)=\{f \in C^k(U): D^\alpha f \text{ is uniformly continuous on ...
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59 views

Compatible maps and Sub- compatible maps

Definition Let $M$ be a subset of a metric space $X$. The set $M$ is called $q$- starshaped if $ kx+(1-k)q \in M$ for all $x\in M$ and $k\in[0,1]$, where $q$ is an element of $M$. Definition Let $M$ ...
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79 views

$f_n \rightharpoonup f$ in $L^q(Q)$ $\forall q < \infty$ and $f_n' \rightharpoonup f'$ in $L^2(0,T;H^{-1})$ implies $f_n \to f$

(... in $C^0([0,T]; H^{-1})$. ) Let $f_n$ be a sequence of functions defined on $Q:=(0,T)\times \Omega$, where $\Omega$ is a bounded domain. I have read this: Since $f_n \rightharpoonup f$ in ...
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108 views

Spectral decomposition of $TT^*$

On $l_{2}$ let $T$ be given by $Te_{n}=\frac{e_{n+1}}{n+1}$ where $(e_{n})_{n\ge1}$ is the canonical orthonormal basis. Find the spectral decomposition of $TT^*$. I find that ...
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102 views

Two possible definitions of “vector-valued distribution”

Let $X$ be a reflexive Banach space. Define $$\tag{1} \mathcal{D}^\star(0, T; X)=\left\{ u\colon \mathcal{D}(0, T)\to X\ \text{linear and continuous}\right\} $$ where the topology on the space of ...
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40 views

How do they call the topological tensor product that classifies operators from Hilbert space?

Let $V$ and $W$ be topological vector spaces. There are different ways to complete the tensor product $V \otimes W$, and the only ones that are usually discussed in introductory literature are the ...
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88 views

Ultraweak topology on Banach spaces

If $X$ and $Y$ are Banach spaces with $Y$ reflexive, then the space $\mathcal{B}(X,Y)$ of bounded operators from $X$ to $Y$ is the dual of the projective tensor product of $X$ and $Y^{*}$. As in the ...
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124 views

norm of differential operator on $P^n[0,1]$

Consider the space $P^n[0,1]$ of polynomials of degree $\leq n$ on $[0,1]$, equipped with the sup norm. Now, this is a finite dimensional space, so all linear operators have to be continuous, hence ...
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66 views

Shift and ergodic measures

Let $X = \{0,1\}^{\mathbb{N}}$ endowed with the product topology and $\sigma : \left\{ \begin{array}{ccc} X & \to & X \\ (\epsilon_n) & \mapsto & (\epsilon_{n+1}) \end{array} \right.$. ...
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78 views

Linear dimension of banach spaces

Let $X$ be some vector space (over $\mathbb{C}$). Note that if $X$ is of finite dimension we can identify $X$ with $\mathbb{C}^n$ for some natural $n$ and endow it with a norm $||x||=|x_1|+...+|x_n|$. ...
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111 views

If $u$ has a weak derivative and $f$ is $C^1$ does $fu$ have a weak derivative (fractional Sobolev space and weak time derivatives)

Let $\Omega$ be an open bounded set. Let $s \in (0,1)$ and $H^s(\Omega) := W^{s,2}(\Omega).$ Let $f \in C^1([0,T]\times \Omega)$ and $u \in L^2(0,T;H^s(\Omega))$ with weak derivative $u' \in ...
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86 views

Regularity theorem for Laplacian

Let $\Omega \subset \mathbb R^d$ be a bounded domain, $d>2$. Let $f \in C^\infty(\Omega)$. If $u \in L^2$ is a distributional solution of $\Delta u = f$ in $\Omega$ then $u \in C^\infty(\Omega)$ ...
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47 views

How can I show that given a norm one linear functional on $c_0$ that there is a unique extension to a norm one functional on $\ell_\infty$?

We are given that our Banach space is $c_0 \subset \ell_\infty(\mathbb{N})$ and there is a functional $y^* \in c_0^*$ such that $||y^*|| = 1$. We are guaranteed that this extends, via Hahn-Banach to a ...
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112 views

Can the “inducing” vector norm be deduced or “recovered” from an induced norm?

Can the "inducing" vector norm be deduced or "recovered" from an induced (operator) norm? This question occurred to me after seeing this question. I'm hoping that perhaps there exists something like ...
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124 views

Non linear compact map

Suppose to have two Banach spaces $E$ and $F$, with $E$ reflexive. Suppose to have a continuous map $T:E \to F$ which maps bounded subsets into precompact subsets. $T$ is not assumed to be linear. ...
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113 views

Showing a sequence $x_{n+1} = Tx_n$ forms a contraction mapping

I want to show that the sequence given by $$x_{n+1} = Tx_n = x_n-\frac{(x_n^2-2)}{x_n+x_{n-1}}$$ forms a contraction mapping. That is $$|Tx_1-Tx_2|\leq c|x_1-x_2|.$$ Where $c$ is to be determined. I ...
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100 views

For $A$ self-adjoint, $\sup_{|x|=1}\langle Ax,x\rangle = \max \sigma(A)$

For a self-adjoint operator $A$ on a Hilbert space $H$, one has $\sup_{|x|=1}\langle Ax,x \rangle = \max\sigma(A)$. I want to prove this using the spectral theorem. My idea is: Let $a = ...
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118 views

Importance of Schwartz kernel theorem

I am currently reading the proof of the Schwartz Kernel Theorem from Hormander Vol I. At the risk of sounding naive, what is the importance of Schwartz kernel theorem? What are certain insights that ...
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128 views

Show that the following $u\in L^{\infty}\cap H^1(B)$ is a weak solution to the given system.

Let $B=B_{\exp(-2)}\subset\mathbb{R}^2$. I would like to show that a weak solution to the following system (in $B$): \begin{align*} \triangle u_1&=-2|Du|^2(u_1+u_2)/(1+|u|^2)\\ \triangle ...
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154 views

Non strictly singular operators

Let $X$ be a separable Banach space and let $T:X\to X$ be a bounded operator that is not strictly singular. Can we always find an infinite dimensional, closed, and complemented subspace $Y$ of $X$ ...
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159 views

Application of Schwartz Kernel Theorem to Quantum Mechanics

I am currently reading Quantum Mechanics for Mathematicians and have a question about a statement made in the book: Remark. By the Schwartz kernel theorem, the operator B can be represented by an ...
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78 views

Polynomials dense in $C^\infty(U, V)$?

Let $V$ be a Banach space and $U$ an open subset of $\mathbb{R}^n$. Let $C^\infty(U, V)$ denote the Fréchet space of $V$-valued smooth functions on $U$ with seminorms $$ \| u\|_{\alpha,K} = \sup_{y ...
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90 views

An element of $\ell^2$ wanted

I am looking for an element $x=(x_0,x_1,x_2,\cdots)$ in $\ell^2$ such that the sequence $z_n, n=0,1,2,\cdots$ defined by $$z_n=2^n(x_n, x_{n+1},\cdots)$$ is dense in $\ell^2$. It seems that this is ...
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615 views

Strict Inequality in Rudin's Proof of the Riesz Representation Theorem

In Rudin's proof of the Riesz Representation Theorem (step ten), he proves that $$\Lambda h_i \leq \mu(V_i) < \mu(E_i) + \epsilon/n , \quad \mu(K) \leq \sum\limits_{1 \leq i \leq n} \Lambda h_i.$$ ...
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127 views

Is this projection operator onto a subspace of a Hilbert space bounded?

(I copy and paste and edit from Is this operator bounded? Hilbert space projection, my question is almost the same) Let $V \subset H$ be Hilbert spaces (different inner products) with $V$ dense and ...
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84 views

There exists unique $g$ s.t. $g(x) = f(x) + A\int_0^1\sin(x-y)g(y)dy$

I'm doing past papers for a first course on functional analysis. We are not allowed to assume any results from real analysis or topology, so I was surprised to find an exam question, where I couldn't ...
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61 views

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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92 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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82 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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32 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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57 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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278 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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154 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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52 views

A normed space is not separable if and only if it contains an uncountable set of disjoint balls of the same radius

I want to show normed space $E$ is not separable iff $E$ contains uncountable set of pairwise non-intersecting balls of radius $r > 0$. Use contraposive: first prove $E$ is separable then ...
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197 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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77 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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72 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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56 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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92 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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120 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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62 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) = ...
4
votes
0answers
69 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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votes
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74 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 $T^t f(x) := \begin{cases} f(x-t), & t\le x\\ 0, & \text{otherwise} \end{cases}$ ...
4
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85 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 ...