Tagged Questions
1
vote
0answers
22 views
Hahn Banach to get linear functional bounded by sub/superlinear functionals
I am working in a real vector space $V$. I have seen it written that if I have a sublinear functional $p$ and a superlinear functional $q$ such that $q \le p$ then there exists some linear functional ...
1
vote
0answers
27 views
To prove that an operator is bounded [duplicate]
I have this problem:
Let $(H, \langle\cdot,\cdot\rangle)$ a Hilbert space on $\mathbb{C}$
and $A:H\rightarrow H$ a linear operator such that $$\langle A(x),
y\rangle\ =\ \langle x, ...
2
votes
1answer
36 views
Is it possible to write any bounded continuous function as a uniform limit of smooth functions
Is $C^\infty(\mathbb{R})\subset C_b(\mathbb{R})$ dense? I.e. is any continuous bounded function $f:\mathbb{R}\to\mathbb{R}$ the uniform limit of smooth functions?
On any bounded interval this is ...
0
votes
0answers
17 views
Question on a third-order boundary value problems
This is the corollary $2.1$, from the article "Positive solutions of third order semipositone boundary value problems"
if $$u'''=\lambda \left(\sum_{i=1}^m c_i(t)u^{\mu_i}-d(t)\right)+e(t), t\in ...
3
votes
2answers
61 views
Proof that an embedding into $\ell^1$ is compact
Prove that any sequence $(x^{(n)})_{n\in\mathbb{N}}\subseteq\ell^1$ such that $\sum_{k=1}^\infty k\lvert x_k^{(n)}\lvert\leq1$ for all $n\in\mathbb{N}$ has a convergent subsequence.
My thoughts ...
1
vote
0answers
29 views
Why is a *-homomorphism isometric, if it maps strictly positive elements to strictly positive elements?
I have the following exercise:
Let $\pi:\mathcal A \rightarrow \mathcal B$ be a *-homomorphism between two unital $C^*$ algebras $\mathcal A$ and $\mathcal B$ which maps the unit to the unit. Assume ...
1
vote
3answers
90 views
$\|f*g\|_q\leq \|g\|_q \|f\|_1$ and $\|f*g\|_\infty\leq \|g\|_q \|f\|_{q^{'}}$, $(1/q+1/q^{'}=1)$?
Now I'm reading the Young inequality. It says that if $f \in L^p(R)$, $g \in L^q(R)$, $1\leq p,q\leq \infty$, $1/p+1/q\geq 1$. Then how could we have the following inequalities:
$$\|f*g\|_q\leq ...
1
vote
1answer
42 views
Existance of function of norm one on normed spaces
On a past exam in a course on functional analysis the following problem is given:
Let $X,Y$ be normed spaces and let $x\in X$ be nonzero. Show that there exist some $f\in X^\ast$ s.t. $f(x) = \|x\|$ ...
3
votes
1answer
43 views
The Subspace $M=\{f\in C[0,1]:f(0)=0\}$
Let $C[0,1]$ with the supremum norm. It's easy to see that $M=\{f\in C[0,1]:f(0)=0\}$ is a closed subspace and so $C[0,1]/M$ is a Banach space. But I'm having trouble in finding a Banach space ...
2
votes
1answer
65 views
Inequality for norms
Let g(x, y) be function on $X\times Y$. Show that for all $p\geq q$
$$
\|\,\|g\|_{L^q(Y)}\,\|_{L^p(X)}\leq \|\,\|g\|_{L^p(X)}\,\|_{L^q(Y)}
$$
Thsnk you.
0
votes
0answers
32 views
Question on Coordenate Base of a Infinite Banach Space [duplicate]
Let $X$ be a infinite dimensional Banach space. If $\mathcal{B}=\{e_i:i\in I\}$ is a Hamel base for $X$. How to show that only a finite number of the coordinate functionals $e^{*}_i$ will be ...
0
votes
0answers
35 views
Green's function
Please can someone told me how to find the Green's function $G(t,x)$ of BVP : $$u'''(t)=0 , \quad t\in (0,1)$$ and BC : $$u(0)=u'(p)=\int_q^1 w(s)u''(s) ds =0 $$ where $\frac12 < p<q<1$ are ...
1
vote
0answers
32 views
Inequality with $\|\cdot\|_p$ norm
Let $x_1, \ldots, x_{2m}$ be $\{0,1\}$ Bernoulli random variables, i.e. variables which takes values $0$ and $1$ with equal probability. Let $S_m$ be group of all permutations $\pi$ on $\{1, \ldots, ...
1
vote
1answer
71 views
Newton-Raphson in $\mathbb{R}^n$
Let $U\in\mathbb{R}^n$ be open and $f:U\to\mathbb{R}^n$ be a $\mathcal{C}^1$ map. $\exists p\in U$ such that $\;f(p) = 0$ and $Df_{|p}$ is invertible. Define $\phi(x)= x-(Df_{|x})^{-1}f(x)$, show ...
1
vote
1answer
50 views
What are some non-constant functions in the following sets? Are these sets a subspace of $Y = C^1[a,b]$?
Let $Y = C^1[a,b]$. What are some non-constant functions in the following sets? Are these sets a subspace of $Y = C^1[a,b]$.
a) $D = \{ y \in Y: y'(a) = 0,\, y(b) = 1\} $
b) $D = \{y \in Y : ...
5
votes
0answers
125 views
Operator completly continuous
For $\lambda>0$, let $v(t)=\lambda \phi(t)$. Consider the BVP
consisting of the equation $$u'''=\lambda[f(t,[u-v]^*+\gamma)+M(t)] ,t\in (0,1)$$
and (BC):$u(0)=u'(p)=\int_q^1 w(s)u''(s)=0 ...
1
vote
1answer
52 views
exercise: limit orthonormal sequence, “Banach Space Theory”
I have an exercise from "Banach Space Theory":
Suppose $\{x^k\}_{k=1}^\infty$ is an orthonormal sequence in $l_2$, where $x^k:=(x_i^k)$. Show that $\lim_{k\rightarrow \infty} x_i^k =0 \ \forall_{i\in ...
0
votes
1answer
62 views
Examples on the dimension of vector spaces of real functions
Let $S$ be a vector space of functions from $\mathbb{R}^n$ to $\mathbb{R}$, say $S := \{ f:\mathbb{R}^n \rightarrow \mathbb{R} \}$.
I am looking for some examples in which the dimension of $S$ is ...
0
votes
1answer
26 views
if $Y$ and $Z$ be two closed subspace of banach space $X$ .Prove $p:X \mapsto Y$ is continuous.
Here Y and Z are closed subspace of such that $Y\cap Z=\{0\}$and $X= Y+Z$ .I have to prove that $p:X \mapsto Y$ is continuous.where $p(y+z)=y $ $\forall y\in Y$ and$\forall z\in z$. ...
1
vote
0answers
26 views
Find spectrum of the operator and an explicit form for the solution
Let $A: L^2[0,2\pi] \to L^2[0,2\pi]$ be defined by $Au=v$ if
$v(x)=\int^{2\pi}_{0} cos(x+t)u(t)dt$
(a) Find the point spectrum, eigenspace, residual spectrum, continuous spectrum, and spectrum
(b) ...
1
vote
0answers
33 views
Find spectrum of the operator
I need some help to find point spectrum, residual spectrum, continuous spectrum and for this problem.
Let $0 \in\mathbb{C}$. On the complex space $L^2[-1,1]$ consider the operator
$Au=v,$ where ...
2
votes
1answer
52 views
Second annihilator of subspace is the subspace itself?
Let $X$ be a Banach space over $\mathbb{C}$ with dual space $X'$ and let $M,N$ be subspaces of $X',X$ respectively. Define the annihilator subspaces of $M$ and $N$ as
$$
M_\circ = \{x \in X: f(x) = ...
0
votes
1answer
44 views
Operator Norm of a Linear Transformation
PROBLEM For the linear transformation $T:\mathbb{R}^n\rightarrow\mathbb{R}^m$ equipped with the $l^1$-norm, namely, for $x\in\mathbb{R}^n$, $||x||=\sum_{j=1}^n |x_j|$ and similarly for ...
3
votes
1answer
186 views
Question about theorem 3.2 from Morse theory by Milnor
THe demonstration of the theorem 3.2 in the book Morse theory by Milnor
is given in the special case whene the manifold is the Torus ,
My question is : can i prove it in the case where the ...
2
votes
1answer
127 views
An other question about Theorem 3.1 from Morse theory by Milnor
In the proof of theorem 3.1 they put : $\langle X,\nabla f \rangle =X(f)$ after that they say that:
for a curve $c$ on $M$ then $\left\langle\dfrac{\mathrm{d}c}{\mathrm{d}t},\nabla ...
1
vote
3answers
95 views
Continuous Function on a Closed Bounded Set in $\mathbb{R}^n$ then that function is bounded and uniformly continuous
Theorem : Let $A$ be closed bounded set in $\mathbb{R}^n$, and let $f:A\rightarrow\mathbb{R}$ be continuous. then $f$ is bounded and uniformly continuous on $A$.
I've been proved this theorem, my ...
2
votes
2answers
50 views
little question about linear operators
Let H be a complex Hilbert Space. Let $P \in L(H)$ be an idempotent operator ($P^{2} = P$). Also, let $\parallel P\parallel = 1$. I want to prove that $P$ is an orthogonal operator. I defined $M = ...
1
vote
0answers
75 views
Are continuous functions strongly measurable?
Measure theory is still quite new to me, and I'm a bit confused about the following.
Suppose we have a continuous function $f: I \rightarrow X$, where $I \subset \mathbb{R}$ is a closed interval and ...
1
vote
1answer
52 views
Orthogonality & Adjoint Operator
I am trying to prove this simple statement left to the reader in Brézis's book.
Let $A \colon D(A)\subset E \longrightarrow F$ be an unbounded operator. Let $G:=\operatorname{Graph}(A)$ and $L=E ...
1
vote
0answers
41 views
best approximation property of finite dimensional banach space.
recently i am stuck in a question whose partly answer is known to me. But i want full answer. Can anyone help me?
The question is following:
Suppose $X$ is a finite dimensional normed linear space ...
2
votes
0answers
49 views
Typical problem in functional analysis #2
I need help with this problem from my homework
Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ and let $\kappa : \Omega\rightarrow\mathbb{R}$ be a continuous function, such that there's ...
4
votes
2answers
60 views
Examples of Banach spaces
Which of the following are Banach spaces?
A. The set of all real-valued functions $f$, $g$ which are functions of an independent real variable $t$ and are defined and continuous on the closed ...
1
vote
1answer
63 views
Typical problem in functional analysis
I need help with this problem from my homework
Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ and let $\kappa : \Omega\rightarrow\mathbb{R}$ be a continuous function, such that there's ...
4
votes
1answer
52 views
Normed vector spaces and Banach spaces
Let $X$ be a Banach space with norm $||.||$ and let $S$ be a non-empty subset in $X$. Let $F_b(S,X)$ be the vector space of $F(S,X)$ of all functions $f:S \rightarrow X$ such that $\{||f(s)||:s \in ...
3
votes
2answers
62 views
Banach spaces and quotient space
Let $X$ be a normed vector space, $M$ a closed subspace of $X$ such that $M$ and $X/M$ are Banach spaces.
Any hint to prove that $X$ must be a Banach space?
-1
votes
0answers
45 views
Linear operator on normed spaces
Let $$g_{t}(x)=\frac{1}{\left( 4\pi t\right) ^{\frac{n}{2}}}e^{-\frac{x^{2}%
}{4t}},\ t > 0,\ x \in \mathbb{R},$$ be the Gauss-Weierstrass kernel.
Prove that the operator
$$T_t : L_p(\mathbb{R}) ...
2
votes
1answer
69 views
Matrix Representation of Trace Class Operators
Suppose we have a separable Hilbert space (thus with a countable basis) and that represent an operator in matrix form, i.e:
$A: H \rightarrow H $$$x \;\rightarrow \sum_{j \in \mathbb{N}}\left(\sum_{k ...
1
vote
1answer
49 views
Uniformly Integrable of sets in $L_{1}(\mu)$ is equivalent to almost order boundedness
A bounded set $F\in L_{1}(\mu)$ is said to be uniformly integrable if : $\forall \epsilon$ there is a $\delta>0 $, such that $\forall$ measurable set $A$, and $\forall f\in F$ , if $\mu(A)< ...
2
votes
0answers
111 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 ...
5
votes
1answer
41 views
States on a C*algebra
A state $\omega$ on a unital $C^*$ algebra $A$ is called definite at $a\in A$ self-adjoint if $\omega(a^2)=\omega(a)^2$.
I proved that if we have such a definite state at $a$, then for all $b\in A$ ...
2
votes
1answer
60 views
On the Spectral Theorem
Let $H$ be a Hilbert space, $T\in B(H)$ be normal and $E$ its spectral measure.
a- Let $\delta >0$ , and let $M_{\delta}$ = $\left\{\lambda\in \sigma(T): |\lambda|\geq \delta\right\}$. ...
6
votes
1answer
128 views
Maximal abelian subalgebra of Banach algebra is closed and contains the unity
I'm studying Murphy's book: C*-Algebras and Operator Theory, and got stuck in exercise 8 from chapter 1:
"Show that if $B$ is a maximal abelian subalgebra of a unital Banach algebra $A$, then $B$ is ...
2
votes
2answers
63 views
Compact inclusion in $L^p$
Is it true that there is a compact inclusion from $L^p$ to $L^q$ whith $q<p$?
What is the counterexample if what I said is wrong?
Thank you.
2
votes
1answer
97 views
Matrix Representation of Operators in Infinite Dimensional (Separable) Hilbert Spaces
Suppose we have a separable Hilbert space (thus with a countable basis) and that we to represent an operator in matrix form, i.e:
$$A: H \rightarrow H \\ \; \; \; \; \; \;x \;\rightarrow \sum_{j \in ...
3
votes
2answers
98 views
Unitary operator in von Neumann algebra
Let $R\subseteq B(H)$ be a von Neumann algebra, and $U\in R$ be unitary. Prove that there is a self adjoint operator $A\in R$ such that $||A||\leq \pi$, and $U=\exp(iA)$ . Any idea how to start! Thank ...
1
vote
0answers
26 views
Exchanging LimSup and Sup
Let $m$ be a probability measure on $W \subseteq \mathbb{R}^p$.
Consider a bounded upper semicontinuous function $f: X \times Y \rightarrow \mathbb{R}_{\geq 0}$, where $X \subseteq \mathbb{R}^n$ is ...
0
votes
0answers
20 views
$\sup$ of Integral preserves Upper Semi-Continuity?
Let $m$ be a probability measure on $W \subseteq \mathbb{R}^p$, so that $m(W) = 1$.
Consider a function $f : X \times Y \times W \rightarrow [0,1]$, where $X \subset \mathbb{R}^n$, and $Y \subset ...
1
vote
1answer
24 views
Prove that $ A\subset \ell_1 $ is compact iff $A$ satisfies the following property
$A$ is compact iff $ A $ is bounded and, given $\epsilon > 0$, there exists $ n_0$ such that $ \sum_ {k=n}^\infty |x_k|\le\epsilon $ for all $n \geq n_0 $ and for all $ x\in A $.
To prove ...
4
votes
1answer
132 views
Using Lax Milgram to find a weak solution in an intersection of Sobolev spaces
I am trying to prove the existence of a weak solution of the problem:
$$
-\Delta^2 u = f \in L^2(U)\\ \\
u|_{\partial U}=\Delta u|_{\partial U} = 0
$$
on the bounded open set $U\subset\mathbb{R}^n$ ...
4
votes
1answer
113 views
Existence of a non zero element in the dual
Let $S$ a vector subspace of a normed vector space $X$ such that $\overline{S} \neq X$. Show that, with the Hahn-Banach Theorem (Geometric Version), that there is $F\in X^{\prime}$ such that ...
