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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Difference isometrically isomorphism and homeomophism.

What is the difference between isometrically isomorphism and homeomorphism?is an isometric mapping is continuous?
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can we expect a bounded approximate identity in $L^{1}(\mathbb R)$, whose Fourier transform at particular point is zero?

Let $\phi:\mathbb R \to \mathbb C$ be a function; and define $$\phi_{t}(x):=\frac{1}{t}\phi (x/t), (t>0).$$ We let, $\phi\in L^{1}(\mathbb R),$ then $\phi_{t}\in L^{1}(\mathbb R);$ in fact, we ...
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18 views

Show that P is an L-projection iff $P^{*}$ is an M-projection

I have started reading "M-ideals in Banach spaces and Banach algebras", but I stuck on the first page. It says that "there is an obvious duality between L- and M- projections: P is an L-projection ...
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37 views

Soft Question: Idea behind complete orthogonal system [on hold]

this is my foray into fourier analysis and I'm already daunted by some of the terminology used. I hope some of you can provide me with some answers that are intuitive and approachable. What is a ...
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1answer
21 views

Paley projection is not absolutely summing

It is well known that the operator $P: H^{1}(\mathbb{T}) \to \ell_2$, given by $Pf= (\hat{f}(2^{n}))_{n \in \mathbb{N}}$ is bounded and its restriction $P_{|A(\mathbb{D})}: A(\mathbb{D}) \to \ell_2$ ...
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63 views

Alternative proof for the fact that a continuous function on a closed interval attains its boundaries.

Let $f:[a,b]\to \mathbb{R}$ be a continuous function. We are interested in showing that $\exists \beta \in [a,b]$, such that $f(\beta) = M$, where M is its upper boundary. I have managed to proof ...
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1answer
23 views

Trace theorem for Sobolev functions: what is the significance of continuous extension to the boundary?

Why in the proof of the trace theorem in L.C.Evans' book on PDE, he has considered functions in $W^{1,p}(\Omega)\bigcap C(\overline{\Omega})$ when it is enough to choose functions in ...
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24 views

Fredholm index for 1-d Schroedinger operator

if I look at a Schroedinger-operator $-\frac{d^2}{dx^2} +V$ on a compact intervall $[a,b] \subset \mathbb{R}$ and I take boundary conditions that this operator is self-adjoint (for example periodic ...
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2answers
34 views

Introduction to Toeplitz operators

I just finished my undergraduate education in mathematics, and i'm starting a graduate program, and i get interest for learning to work with Toeplitz operators, but i have no background with ...
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1answer
37 views

Proof of the Hahn-Banach separation theorem

In the proof of the Hahn-Banach separation theorem my notes claim the following: Let $X$ be a normed $\mathbf{R}$ vector space, $A,B\subset X$ be nonempty, disjoint, convex, $A$ compact and $B$ ...
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16 views

Simple question about matrices cones

Fix a dimension $d>1$ here. A cone is non empty closed and convex subset $K$ of $\mathbb{R}^{d}$ such that $tv\in K$ whenever $t>0$ and $v\in K$ and $K\cap -K=\{0\}.$ I have a finite set of ...
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24 views

TT* Duality argument

I am trying to obtain $L^p$ estimate for a system of nonlinear PDE. I do not have $L^2$ to work with for my problem, I heard $TT^*$ argument is useful tool if one wishes to mapp from $L^p$ to $L^p$. ...
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1answer
10 views

Question about Neighborhood basis

In the Simon Reed text, after defining the strong operator topology it is said: "A neighborhood basis at the origin is given by the sets of the form $\{S \ | \ S \in \mathcal{L}(X,Y), ...
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14 views

Basis representation for non-negative, compact support, reasonably smooth spectral function

I was wondering if anyone has ideas on representing a non-negative, compact support (from x=-1 to 1 on the real axis) spectral function as a superposition of basis elements. Ideally, the basis ...
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30 views

Exercise about spectrum of selfadjoint operator.

I'm stuck on an exercise about the spectrum of a selfadjoint operator on a Hilbert space. The problem is the following: Let $(X,\langle \cdot, \cdot\rangle)$ a Hilbert space and let $A \in B(H)$ a ...
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34 views

Operator $Au(t) = \int_0^t e^{t-s} u(s) ds$ (Proof Verification)

Consider the space $C([0,1])$ with $||\cdot||_\infty$ norm. Let $A: C([0,1])\rightarrow C([0,1])$ be the operator defined by $$Au(t) = \int_0^t e^{t-s} u(s) ds.$$ And I am not 100% sure about (c), ...
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26 views

Finite Measure Space: Integral Closure = Bochner Integral

I can't sleep for so long time as the integral gives me headaches. I was looking for so many approaches. Now another one. Hope this works... Let $\Omega$ be a finite measure space and $E$ a Banach ...
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86 views

Two questions in spectral theory: the spectrum of the Fourier transform and the Hamiltonian of the hydrogen atom.

I have the following two questions: The Fourier transform defines a unitary (provided that it is normalized properly) map $\hat{\cdot}:L^2(\mathbf{R})\rightarrow L^2(\mathbf{R})$. I figured out its ...
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1answer
44 views

Properties of the Double Layer Potential

Consider the double layer potential $$ W_{\nu}(x) = \int_{\partial\Omega} \nu(y) \frac{\partial}{\partial n_y}\left( \frac{1}{|x - y|} \right) d \sigma_y $$ for a bounded region $\Omega \in \mathbb ...
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1answer
32 views

How do the inner products on $L^2$ look like?

I was wondering whether all scalar products in $(L^2[0,1],\lambda)$ are given by $\langle f,g \rangle := \int f(x)g(x) \cdot w(x) d\lambda(x)$? If this is true, what are the exact conditions that we ...
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43 views

Functions with compact support

I have a question about a convergence of functions with compact support. SETTING Let $d\geq 3$ and $U \subset \mathbb{R^{d}}$ be open and $dx$= Lebesgue measure on $U$. Let $b_{i},c,d_{i} \in ...
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1answer
14 views

Lower semicontinuity

Let $\Omega\subset\mathbb R^n$ be open and bounded. I consider a sequence $u_k:\Omega\to\mathbb R$ of smooth functions which converges uniformly to a function $u:\Omega\to\mathbb R$. Moreover, the ...
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22 views

L1 error for scale/translation classes

This is an example given in the article about testability (Devroye and Lugosi 1990) (link: http://repositori.upf.edu/bitstream/handle/10230/1024/375.pdf?sequence=1 ) page 7. First I will introduce my ...
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29 views

Properties of the essential supremum

In the article about testability of densities (Devroye and Lugosi 1999, page 4) (link: http://repositori.upf.edu/bitstream/handle/10230/1024/375.pdf?sequence=1 ). They use some properties of the ...
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1answer
16 views

About Sobolev-Poincare inequality on compact manifolds

Let $2^* = \frac{2n}{n-2}$ where $n$ is the dimension of a compact closed manifold $M$. We get from the Sobolev/Poincare inequality the identity $$\lVert u \rVert_{2^*} \leq C\lVert \nabla u ...
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32 views

Limit of integral of L^p functions

Let $p\in (0,\infty)$ and $f\in L^p(\mathbb{R})$. Show that $\displaystyle \lim_{n\to\infty} \int_{\mathbb{R}} f(x) \chi_{[-n,n]}\frac{1}{n^{(1-1/p)}} dx=0$. I believe $f(x) ...
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1answer
16 views

Why does T symmetric imply T* extends T?

This is a result I've seen stated a few times, but I can't seem to come up with a proof! Suppose $T$ is a densely defined linear operator with domain $D(T)\subset H$, where $H$ is a Hilbert space ...
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1answer
18 views

Showing that a function is bounded in $L^1$ given a bound on its distribution function

Let $f \in L^2((0,T)\times\Omega)$ where $\Omega$ is a compact manifold. Suppose I know that for every $k > 0$, $$\mu(\{|f| > k\}) \leq Mk^{-\frac 12}$$ for some constant $M$ (which is ...
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59 views

How to show a set is compact in a function space?

I have a question asking if $\{f_n\}$ is a compact in $C_b([0,\infty))$ (bounded continuous) with $||\cdot||_{L^\infty}$. The sequence is $$f_n (t) = \sin\sqrt{t+(2n\pi)^2},$$ I have showed that ...
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1answer
28 views

If $S$ is dense in $L^{2}$. Is it true that $pS=\{pf| f\in S, pf\in C^{\infty}\} $ is dense?

Let $S=\{f\}$ be a set of function defined in a compact subset $\Omega\subset \mathbb{R}^{n}$ such that $S$ is dense in $L^{2}(\Omega)$. Is it true that for $p\neq 0$ a rational function $pS=\{pf| ...
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34 views

Contraction mapping principle application

I'm to prove that the following equation has a unique solution: $$f(x) = \int_0^1 e^{-sx} \cos(\alpha f(s)) ds.$$ (Here, $\alpha \in (0,1)$.) The form of the exercise screams to apply the ...
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1answer
19 views

Contraction mapping problem

Let $T$ be the following operator on $C[0,1]$: $$(Tu)(t) = u(0) + \lambda\int_0^t u(\tau)d\tau$$ where $\lambda \in (-1,1) \subset \mathbb{R}$. Then I need to show $T$ is a contraction. So I need ...
2
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1answer
23 views

Definition and analyticity of $T^z$ where $T$ is a positive operator

Let $H$ be a Hilbert space. Suppose that $T\colon D(T) \to H$ is a positive selfadjoint operator where $D(T)$ is the domain of $T$. The spectrum $\sigma(T)$ of the operator $T$ is a subset of ...
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1answer
63 views

A special property of $\limsup$ in $\ell_1$

Let $w$ be any element in $\ell_1$, and $(w_n)$ be a bounded sequence in $\ell_1$ that converge to 0 pointwise. I want to prove ...
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23 views

a (probably trivial) question on trace theory

I was going through the trace theorem on Sobolev space which speaks of its existence for function $\in W^{k,p}(\Omega)$. My question is whether the trace operator unique if it exists?. My intuitive ...
3
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1answer
28 views

Local convexity of the topology of weak convergence of probability measures

Let $X$ be a Polish or standard Borel space, and $\mathcal P(X)$ be the space of all Borel probability measures on $X$ endowed with the topology of weak convergence. I am thinking of using ...
2
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1answer
67 views

Does a function that is twice weakly differentiable have a version that is classically differentiable?

I have been wondering about the idea of functions that are weakly differentiable. My intuition tells me that the weak derivative allows one to differentiate functions that either have a removable ...
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69 views

$C(X)$ is separable when $X$ is compact

Let $X$ be a compact space and let $\Bbb U =\{(U,V); U,V \mbox{ are open subsets of }X \mbox{ and }\mathrm{cl} U \subset V\} $. for $u=(U,V)$ in $\Bbb U$ , let $F_u:X\to [0,1]$ be a continuous ...
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1answer
24 views

Linear operator defined by its eigenvectors/values

Let $H$ be a Hilbert space, $(e_n)$ a complete orthonormal sequence, and $\lambda_n$ a bounded sequence of complex numbers. Let $A$ be defined such that the $(e_n)$ are the eigenvectors of $A$ and the ...
2
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1answer
27 views

Continuity of a functional with respect to two different norms

Let $J$ be a functional defined on $E = C^1[a,b]$ by $$J(y) = \int_a^b \sqrt{1 + (y^{\prime}(x))^2} \, dx.$$ Define the following two norms on $E$: $$\|y\|_{\infty} = \max_{a\leq x\leq b} |y(x)|$$ ...
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23 views

Closest point in a given subspace

Let $X$ be a normed space (not necessarily Banach) and let $\{v_1,... v_n\}$ be a linearly independent subset of $X$. For fixed $y \in X$, I'm to show that there are scalars $a_i$ minimizing ...
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22 views
+50

A question about conic sets of functionals.

The problem is the following. Let $(X,\|\cdot\|)$ be a normed space and let $C \subseteq X$ be a closed convex set with nonempty interior. Let be $x\in C$. I define to be a "normal cone to $C$ in ...
2
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1answer
41 views

Need help with application of Hardy-Littlewood inequality (Marcinkiewicz space and distribution functions)

I am going over this work here. I couldn't understand the equality where the Hardy-Littlewood inequality is used. I think $\delta$ here is a weight so we can take it to be $1$ for simplicity. Would ...
2
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1answer
47 views

Differentiation under the integral if and only if we have an $L^1$ dominator

Let $f(x)\in L^2(\mathbb{R})$ and define $$g(t) = \int_\mathbb{R} f^2(x)\exp(-tx^2)dx$$ for $t\geq0$. We want to show that $g(t)$ is continuously differentiable if and only if $xf(x)\in ...
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1answer
47 views

Relation between Sobolev Space $W^{1,\infty}$ and the Lipschitz class

I have a Sobolev space related question. In the book 'Measure theory and fine properties of functions' by Lawrence Evans. I know the result that states that for $f: \Omega \rightarrow \mathbb{R}$. $f$ ...
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1answer
11 views

Is $\text{Id} = \chi_{\{ |u| \leq k\}} + \chi_{\{|u| > k\}}$ well defined for $u \in L^p(0,T;L^q)$?

Is the decomposition $$\text{Id}(z) = \chi_{\{ |u| \leq k\}}(z) + \chi_{\{|u| > k\}}(z)\tag{1}$$ well defined for $u \in L^p(0,T;L^q(\Omega))$? I guess (1) holds a.e. So the problem is, is the set ...
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45 views

norm of a singular integral operator

My question is from Harmonic Analysis, about the study of singular kernels (in the Calderon Zygmund sense.) Suppose that a kernel $K$ has the cancellation property, and decays like $\frac{1}{|x|^n}$ ...
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1answer
20 views

Integral of a composition of piecewise linear function with polynomial

Fix a number $k > 0$ and let $$T(x) = \begin{cases} k &: x \geq k\\ x &: |x| < k\\ -k &: x \leq -k \end{cases}. $$ Define $S(s) = \int_0^s T(|x|^{m-1}x)\;dx.$ I want to show that ...
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1answer
40 views

Semi-norms in Functional Analysis

I'm self-studying functional analysis. The following is from Rudin's "Functional Analysis, 2nd edition". It consists of parts from question 7 and 13 from the first chapter. I am not sure if my answers ...
2
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46 views

Any other operators that may convert algebraic function into transcendental ones

As we know, the integral may convert or map a rational function or algebraic function into a transcendental one. Are there any other operators that may convert a rational function or algebraic ...