1
vote
0answers
12 views

Show that $J:C(X)\to C(Y)$ is positive.

I've been breaking my head over this one for quite some days now. I hope someone here has better insights then me. Here $X,Y$ are compact Haussdorff-spaces. Then $C(X),C(Y) $are a banach spaces with ...
3
votes
1answer
22 views

Is the set of all Taylor polynomials a vector space?

Let $V$ denote the set of all Taylor polynomials of degree $\leq n$ for a fixed natural number $n$ (including the zero polynomial), regraded as real-valued functions of a real variable. Then is $V$ a ...
0
votes
0answers
15 views

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 ...
6
votes
1answer
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 ...
1
vote
0answers
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 ...
0
votes
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 $B$ ...
1
vote
1answer
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), ...
0
votes
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 ...
3
votes
0answers
44 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 ...
0
votes
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 ...
1
vote
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 ...
3
votes
1answer
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 ...
2
votes
1answer
48 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 ...
0
votes
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 $f: \Omega \rightarrow \mathbb{R}$. $f$ ...
1
vote
1answer
86 views

Spectral theory

I have absolutely no idea about Spectral theory and want to ask some fundamental questions. 1.) What does it mean that the resolvent of an operator is Hilbert-Schmidt? (Cause I saw a theorem that was ...
-1
votes
1answer
30 views

derivative of a smooth compactly supported functions is also compactly supported [on hold]

I would want to know if the following implication is right: Define $ \mathcal{D}(\Omega):=\{\varphi(x)\ |\ \varphi \in C^\infty(\Omega) \text{ and } \varphi \text{ has compact support} \}$ $$ \phi(x) ...
3
votes
1answer
124 views

Monotone Convergence Theorem for Riemann Integrable functions

I'm having a really hard time proving this statement (this is not homework): If $f_{n} : [0,1] \rightarrow \mathbb{R}$ is a Riemann integrable function for all $n \in \mathbb{N}$, and $0 \leq f_{n + ...
2
votes
1answer
45 views

Are these two expression square integrable?

I have two expressions (let's call them functions $f,g$) on $[0,1]$, where I want to find out whether they are square-integrable or better: for which $m \in \mathbb{Z}$ they are square-integrable ( ...
0
votes
0answers
31 views

Convergence of norms

I have this space $H_{0,p}^1=\lbrace u\in AC([0,+\infty),\mathbb{R}),u(0)=u(+\infty)=0, \sqrt{p} u'\in L^2(0,+\infty)\rbrace $ endowed with the norm $||u||^2=\int_0^{+\infty} p(t) u'^2(t) dt$ ...
0
votes
1answer
17 views

$f\in L^{p}(\mathbb R)\cap C_{0}(\mathbb R); (1<p<\infty), g\in C^{\infty}_{c}(\mathbb R) \implies f\ast g \in C^{k}(\mathbb R)$?

We put, $C_{0}(\mathbb R)=$ The space of continuous functions on $\mathbb R$ vanishing at $\infty$; $C^{k}(\mathbb R)=$ The space of all functions $\mathbb R$ whose derivative of order $\leq k$ exist ...
1
vote
1answer
33 views

Proof that this set is not compact

Let $X=C[0,1]$ with the $\sup$ norm. Let $Y = \{f\in X\mid \|f\|_\infty \le 1\}$. It is my goal to show that $Y$ is not compact using the sequence defintion of compactness. Note that it is very easy ...
2
votes
2answers
102 views

How to reduce this to Sturm-Liouville form?

I have the ODE $$-(1-x^2) \frac{d^2 f(x)}{dx^2} + x \frac{df(x)}{dx}+g(x)f(x)=\lambda f(x)$$ and I want to reduce it to Sturm-liouville form. The problem is that we don't have $2x$ but just $x$. ...
2
votes
2answers
70 views

The trace of an operator

My question is derived from A. Deitmar's book: A First Course in Harmonic Analysis (second edition), p22, Exercise 1.17. Let me rewrite it again: Let $k:\mathbb{R}^2 \rightarrow \mathbb{C}$ be smooth ...
1
vote
3answers
38 views

Dense subsets of $(L^p(\Omega),\|\cdot\|_p)$

The following results hold. Theorem Let $\Omega\subset\mathbb{R}^n$ be an open set. Then $C^0_c(\Omega)$ is dense in $(L^p(\Omega),\|\cdot\|_p)$, if $1\le p<\infty$. Theorem Let ...
1
vote
1answer
49 views

$C^0$ is a closed subspace of $L^{\infty}$

Let $\Omega\subset\mathbb{R}^n$ be an open bounded set. Let $f\in C^0(\bar\Omega)$. I have to prove that $\|f\|_{\infty}=\|f\|_{L^{\infty}}$. One implication is trivial. Let's consider the other one. ...
0
votes
0answers
46 views

How to find an example

I want to find a function $f\in C^1([0,+\infty)\times\mathbb{R},\mathbb{R})$ such that $f(t,0)=0$ $f(t,u)\leq \alpha u+\beta$, $\alpha<\lambda_1,\beta\geq 0$ $f(t,u)\geq C_1 |u|^{\sigma}$ where ...
3
votes
1answer
30 views

Classify the continuous bilinear functional on $L^p \times L^q$.

Let $1<p<\infty$, $1/p+1/q=1$ and let $L(\cdot,\cdot)$ be continuous bilinear functional on $L^p(\mathbb{R}) \times L^q(\mathbb{R})$. The continuity means that if $f_{n} \rightarrow f$ in $L^p$ ...
0
votes
2answers
45 views

Representation of linear functionals on a certain Banach space

Let $C^k([0,1])$ be the space of such complex-valued functions on $[0,1]$ that are continuously differentiable at least $k$ times ($k\in\mathbb N$). It is well known that $C^k([0,1])$ is a Banach ...
2
votes
1answer
34 views

Spherical coordinates and Lebesgue integral

I found a textbook that says that $r \in [0,\infty), \theta \in [0,\pi], \phi \in [-\pi,\pi]$ are the spherical coordinates. Then he goes on with: This corresponds to a unitary map $L^2(\mathbb{R}^3) ...
2
votes
1answer
34 views

What is the definition of this set of absolutely continuous function

I know that $$AC(a,b):=\left\{f \in C(a,b)|f(x) = f(c)+\int_c^x g(t) d \lambda(t),c \in (a,b), g \in L^1_{\text{loc}}(a,b)\right\}$$ $$AC[a,b]:=\left\{f \in C[a,b]|f(x) = f(c)+\int_a^x g(t) d ...
2
votes
0answers
63 views

Problem in functional analysis.

I heard of this problem that caught my attention and I am curious now thus I would appreciate if I could have a hint or a solution. Let $(x_n)$ a sequence in a normed space $X$ such that ...
1
vote
1answer
41 views

The existence of $f \in C^\infty(R^n)$ with $ f=0$ on closed $E$, otherwise $f>0$

This is problem 6.3 in 'Rudin's Functional analysis If $E$ is an arbitrary closed subeset of $R^n$, show that there is an $f \in C^\infty(R^n)$ such that $f(x)=0$ for every $x \in E$ and ...
0
votes
0answers
74 views

Functional equation relating to normal numbers

My coauthor and I have run into the following problem in a research project involving normal numbers. We suspect that the following question may be resolved using standard techniques in analysis. We ...
2
votes
1answer
46 views

The dual space of locally integrable function space

I'm strongly interested in dual spaces. I learned the dual spaces of $L^p$, $L^{\infty}$, $C(X)$ and $M(X)$. I wonder the dual space of locally integrable function space in $\mathbb{R}^n$. The ...
2
votes
0answers
39 views

Convergence of Step Function Defined by Averages

For a function $f \in L^2[0,T]$, and a uniform partition $P = \{0=t_0, t_1, \ldots, t_n = T\}$ of the domain, we can define a step function approximation as the average value over each interval in the ...
2
votes
3answers
101 views

Prove that $f_n(x) → f(x)$ uniformly on $E$ as $n → ∞.$

Let $E ⊂ \mathbb{R}$ be a compact (i.e., closed bounded) set of real numbers. Suppose $\{f_n\}$ is a sequence of real-valued continuous functions which converges pointwise on $E$ to a function $f$ ...
0
votes
1answer
124 views

Is this proposition about $L^2$ functions correct?

Is this proposition correct? Will you please give a contour example if it is wrong? If $J \in L^2(\mathbb{R}) \cap C^1 (\mathbb{R})$, $f \in C^{\infty}(\mathbb{R})$, $|f'|\leq K$, where $K > 0$ is ...
1
vote
1answer
33 views

Strong Convergence in L1 Implies Weak Convergence in L2?

If I have $f_n \to f$ in $L^1(D)$, where $D \subset \mathbb{R}$ is compact, is it accurate to say $f_n \rightharpoonup f$ in $L^2(D)$? The argument is as follows: consider a simple function $\phi = ...
1
vote
1answer
32 views

Weakly * continuous definition

What does it mean that $$t\to u(t,\cdot)$$ is waakly* continuous from $[0,T]$ to $L^\infty(\mathbb{R}^d)$? I guess that I have to see $L^\infty(\mathbb{R}^d)$ as the dual of $L^1(\mathbb{R}^d)$ and ...
0
votes
1answer
25 views

Formula for $L^{q}$ norm using $C_{c}^{\infty}$ functions

We put, $L^{p}=L^{p}(\mathbb R), L^{q}=L^{q}(\mathbb R);$ $\frac{1}{p}+\frac{1}{q}=1;$ ($p$ and $q$ are conjugate exponents); and $<f,g> =\int_{\mathbb R} f(x)g(x) dx.$ Fix $g\in L^{q}, ...
1
vote
0answers
29 views

boundedness of $\{\int_E(g f_n)\}$ implies boundedness of $\{f_n\}$

I need some help on this problem: Let $E$ be a measurable set, $1 \le p < \infty$ and $q$ is the conjugate of $p$. Suppose that $\{f_n\}$ is a sequence in $L^p(E)$ such that for each $g \in ...
0
votes
0answers
20 views

Proving that integrator operator of a kernel satisfies a specific peroperty

I am trying to prove that a integrator operator of a kernel satisfy a specific property say $\phi$. By integrator operator for non-negative definite kernel $\mathcal{K}$ I mean $T_{\mathcal{K}}$ such ...
1
vote
1answer
52 views

What does it mean to ''construct a Riemann-Stieltjes integral''?

This question refers to an exercise (1.11c) from Reed & Simon's book on mathematical physics. In the first parts of this problem, the reader is introduced to the notion of functions of bounded ...
0
votes
1answer
43 views

Definition of weak divergence [closed]

Can anyone give me the definition of the divergence of a vector field in the distributional sense?
4
votes
2answers
44 views

p-norm of a function

Let $f\in L^1(\mu)\cap L^\infty(\mu)$. I have proved for any $1<p<\infty$, $f\in L^p(\mu)$, $w(p)=||f||_p$ is continuous w.r.t. $p$, and $\lim_{p\to \infty}||f||_p=||f||_\infty$. Is $w(p)$ ...
0
votes
0answers
118 views

Span of Dirac's delta distributions dense in Hilbert space of $L^2$ functions?

According to Wiki a set of elements of a Hilbert space(B) is a basis for that space if: Orthogonality: Every two different elements of $B$ are orthogonal: $⟨e_k,e_j⟩=0$ for all $k$, $j$ in $B$ with ...
5
votes
2answers
158 views

Prove that $f$ is constant on $[a,b]$

$\displaystyle \int_{a}^{b} f^2(x) \, \mathrm{d}x$ = $\displaystyle \int_{a}^{b} f^4(x) \, \mathrm{d}x$ = $\displaystyle \int_{a}^{b} f^3(x) \, \mathrm{d}x$ And $f$ is continious on $[a,b]$ and ...
2
votes
0answers
38 views

If $\phi\in \mathcal{S}(\mathbb R) $ then $\phi_{t}(x)=\frac{1}{t} \phi(x/t)\in\mathcal{S}(\mathbb R)$?

Let $\phi:\mathbb R \to \mathbb C$ be a function; and define $$\phi_{t}(x):=\frac{1}{t}\phi (x/t), (t>0).$$ We note that, if $\phi\in L^{1}(\mathbb R),$ then $\phi_{t}\in L^{1}(\mathbb R);$ in ...
0
votes
0answers
17 views

Let S be the set of of all step functions on [0, 1] with rational range and rational partition points.

Hello need help with this problem: Let $S$ be the set of of all step functions on $[0, 1]$ with rational range and rational partition points. 1- Show that the closure of $S$ in $L^\infty[0,1]$ ...
0
votes
1answer
35 views

How to deal with discontinuous points when proving that step functions are dense in $PC[a,b]$

This question is a follow-up to my previous question: How does one prove that a space is dense in another under some norm? I figured out a way to solve (part of) the exercise. Given some function ...