0
votes
0answers
5 views

Nonexpansive Affine Operators in Hilbert spaces

Let $H$ be a Hilbert space with real inner product $\langle \cdot, \cdot \rangle : H \times H \rightarrow \mathbb{R}$. Let $T$ be an affine operator. Show that $T$ is nonexpansive, i.e., $\left\| ...
1
vote
0answers
23 views

Invertible iff Bounded below and dense range

Statement: Given a Hilbert space $\mathscr{H}$ and $\mathscr{K}$ and a bounded operator $A \in \mathscr{B}(\mathscr{H}, \mathscr{K})$. Show that $A$ is invertible if and only if $A$ is bounded below ...
0
votes
0answers
17 views

Convex Feasibility problem on Hilbert space

Let $H$ be a Hilbert space with real inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$. Consider a closed convex set $X \subset H$ and let $P_X: H \rightarrow X$ be the ...
-1
votes
0answers
16 views

prove that F is dense in C(X×Y,R)?

Let X,Y be compact metric spaces. Let F= {∑ Ai fi(x) gi(y),fi∈∁(X,R),gi∈∁(Y,R), i from 1 to n }. prove that F is dense in C(X×Y,R) ? please i cant figure it out any help i will be thankful !
0
votes
0answers
17 views

Projection and Pseudocontraction on Hilbert space

Let $H$ be a Hilbert space with real inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$. Consider a closed convex set $X \subset H$ and let $P_X: H \rightarrow X$ be the ...
1
vote
1answer
36 views

The Gaussian Integral

Hi I am trying to calculate the expected value of $$ \mathbb{E}\big[x_i x_j...x_N\big]=\int_{-\infty}^\infty x_ix_jx_k...x_N \exp\bigg({-\sum_{i,j=1}^N\frac{1}{2}x^\top_i A_{ij}x_j}-\sum_{i=1}^Nh_i ...
3
votes
2answers
50 views

The set $S=\{(x,y) \in \mathbb{R}^{n} \times \mathbb{R}^n = \mathbb{R}^{2n} ; x \neq y\}$ is connected if $n \geq 2$.

When n = 1 it is easy to see that is not connected, it just take the split open $ S=U_1 \cup U_2$ such that $U_1 = \{(x,y) \in \mathbb{R}^2 ; x > y\}$ is $U_2 = \{(x,y) \in \mathbb{R}^2 ; x < ...
0
votes
1answer
39 views

Separability of functions with compact support

Let $X$ be a locally compact metric space which is also $\sigma$-compact. Let $C_{c}(X)$ be the continuous functions on $f$ from $X$ to $\mathbb{R}$ with compact support. Is $C_{c}(X)$ separable? My ...
0
votes
1answer
23 views

Property of Projection Operator

Let $p: \mathbb{R}^n \rightarrow X$ be the projection mapping onto $X \subset \mathbb{R}^n$ compact convex. I am wondering if $$ \left( p(x) - p(y) \right)^\top z \leq \left( x -y\right)^\top z $$ ...
0
votes
0answers
52 views

Real Analysis and dynamics

I am looking for a textbook or similar resource that addresses the content in a rigorous graduate course in real analysis(at the level of Rudin/Royden) with the following criteria: No hand waving - ...
3
votes
1answer
46 views

Vector spaces isomorphic, then dual spaces isomorphic

If we know that there is a (topological) isomorphism between two Banach spaces $X,Y$ called $\phi \in L(X,Y)$. Then the appropriate isomorphism between the dual spaces $X',Y'$ is given by $\phi' \in ...
2
votes
1answer
22 views

Is the dual of a complete topological vector space always complete?

Let $X$ be a complete topological vector space (over $\mathbb{C}$ say), and $X'$ its dual with the weak*-topology. Then is $X'$ always complete? You may assume $X$ is locally convex if you like.
0
votes
1answer
24 views

$f, \hat {f} \in L^{p} \cap L^{\infty} \implies f\in A(\mathbb R)$?

$1\leq p \leq \infty $. We put, $$X_{p}= \{f\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R) :\hat{f}\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R)\};$$ and we consider the algebra of Fourier ...
2
votes
1answer
32 views

Equicontinuous family of sequence of functions

We are given a sequence of real valued functions $\{g_n\}$ that are defined and continuous on the unit sphere $S$ and differentiable inside it (except at the boundary of the sphere $S$ Also, it is ...
1
vote
1answer
25 views

Eigenvalues of adjoint for residual spectrum.

Statement: Let $T$ be a bounded operator in a Hilbert space $\mathscr{H}$ Show that if $T-\lambda I$ is not dense in $\mathscr(H)$, then $\overline{\lambda}$ is an eigenvalue of $T^*$. Attempted ...
0
votes
1answer
60 views

Is this construction already a contradiction

Let's say we have $\ell^p$ for $p>2$ and $\ell^2 \subsetneq \ell^p$. Let $U$ be a closed subspace of $\ell^2$. Is it possible that $U$ is isomorphic to $\ell^p$ as a Banach space?- I hardly think ...
1
vote
2answers
32 views

Geometrical Meaning of derivative of complex function

What's the geometrical meaning of f'(z) in complex analysis, as we know in real analysis f'(x) has meaning ie. Slope of curve or gives max/ min. But what does derivative f'(z) has geometrical meaning ...
0
votes
2answers
32 views

showing $\inf \sigma (T) \leq \mu \leq \sup \sigma (T)$, where $\mu \in V(T)$

I am trying to prove the following: Let $H$ be a Hilbert space, and $T\in B(H)$ be a self-adjoint operator. Then for all $\mu \in V(T)$, $\inf \left\{\lambda: \lambda \in \sigma (T) \right\}\leq \mu ...
3
votes
2answers
84 views

Any good, undergraduate level introductions to Functional Analysis?

In my lower division math classes, my instructors referenced functional analysis as essentially the extension of linear algebra to infinite dimensional vector spaces along with some real analysis. As ...
1
vote
1answer
20 views

Integral's limit

Let $X$ be a Banach space and $A$ is a linear bounded operator on $X$. It is well known that for $|\lambda|> \|A\|,$ we have $$\|(\lambda I - A)^{-1}\| \leq \frac{1}{|\lambda|-\|A\|}.$$ Now, let ...
2
votes
1answer
35 views

incorrect proof of Hahn Banach Theorem

What is wrong with the following trivial proof of the Hahn Banach Theorem Hahn Banach Theorem: Let $V$ is a real normed vector space and $U$ a subspace. Then if $\phi : U \rightarrow \mathbb{R}$ is ...
2
votes
1answer
21 views

Fast argument to see that the dual map of a projection is a projection

If $X$ is a Banach space and $U,V$ are closed subspaces, such that $X \cong U \oplus V$, then a continuous linear map $P:X \rightarrow X$ is called a projection if $P|_U =id$ and ker(P)=V. Now we ...
1
vote
0answers
12 views

Proving the Rietz-Fischer Theorem for $p = \infty$

Rietz-Fischer Theorem: Let $E$ be a measurable set and $1 \le p \le \infty$. Then every rapidly Cauchy sequence in $L^p(E)$ converges both with respect to the $p$-norm and pointwise almost everyone ...
1
vote
1answer
35 views

weakly convergent subsequence implies strongly convergent

Statement: Let $X$ be a Banach space If $x_n \rightarrow x$ weakly and every subsequence of $\{x_n\}$ has a strongly convergent subsequence, then $x_n\rightarrow x$ strongly in $X$ Attempt: ?
1
vote
1answer
35 views

Is $l^p$ closed in $L^p$?

Let's assume we have a subspace $X$ of $L^p$ and we know that $X \cong l^p$(this should just mean isomorph no isometry is assumed here). Can we infer from this that $X$ is closed? I have just read a ...
0
votes
1answer
15 views

Holder's Inequality Proof Verification

Wikipedia outlines a nice proof of Holder's Inequality in the link provided. The fifth sentence in the proof reads: Dividing $f$  and $g$ by $\|f \|_p$ and $\|g\|_q$, respectively, we can assume ...
1
vote
1answer
42 views

Basic sequence- what is so special about it?

Let $(x_n)$ be a Schauder basis of a vector space $X$. This means that the $span(x_n)$ is dense in $X$, right? Then wikipedia introduces the notion of a $\textbf{basic sequence} $ when $(x_n)$ is a ...
2
votes
3answers
48 views

Gel'fand representation of a non-unital Banach space: what's wrong with this argument

My argument below is hacked together from pages 5-6 of Davidson's "$C^*$ algebras by example". Theorem: The multiplicative linear functionals on a unital abelian Banach algebra are continuous of ...
0
votes
1answer
39 views

Why is the topology of convergence in measure equivalent to this metric here?

I am currently struggeling with the topic of convergence in measure topologies. Now I read that on the space of measurable function $L^0$ on $[0,1]$ with the Borel sigma algebra and the lebesgue ...
0
votes
1answer
35 views

what are closed sets in $L^{1}(\mathbb R)$?

Consider, $L^{1}(\mathbb R)$= The space of Lebesgue integrable functions on $\mathbb R$; for $f\in L^{1}(\mathbb R),$ we define its norm, by $\|f\|_{L^{1}}=\int_{\mathbb R}|f(x)| dx$; It is well-known ...
0
votes
0answers
32 views

Question on Morse inequalities

I want to understand why: if i have then $(4.1)$ is formal : it means that please help me Thank you EDIT1: $(4.1)$ tel us that $\displaystyle\sum_{q=0}^{\infty} (M_q-\beta_q)t^q=(1+t)Q(t)$ ...
1
vote
1answer
14 views

Suprema and Infima of nonpositive functions

I am trying to get some estimates using the time-dependent infimum and supremum of a function $g(t,x)$. I have the following question. Suppose $g(t,x)\leq0$ for all $x\in\mathbb{R}$ and $t\geq0$. ...
3
votes
1answer
45 views

difference between weak* convergence and convergence

I am trying to prove the following: If $X$ is a finite-dimensional space, then for sequences $\left\{x_n\right\}\subseteq X$ and $\left\{f_n^*\right\}\subseteq X^*$, if there exists an $x\in X^*$ ...
2
votes
1answer
70 views

Why is $-\Delta+1$ an isomorphism between Sobolev space $W^{2,p}$ and $L^p\,$?

Consider the linear elliptic operator $L: W^{2,p}(\mathbb{R}^N)\to L^p(\mathbb{R}^N)$ defined by $Lu=-\Delta u + u$. Can we prove that $L$ is an isomorphism for all $p\geq 1$? It can be proved that it ...
2
votes
0answers
36 views

Essential hypothesis of Fourier Inversion formula

Let $f\in L^{1}(\mathbb R)$ and we define its Fourier transform as follows: $\hat{f}(\xi)=\int_{\mathbb R} f(x)e^{-2\pi i \xi\cdot x} dx, (\xi \in \mathbb R);$ and we define $f^{\vee}(x):=\hat{f}(-x)= ...
0
votes
1answer
25 views

showing that a sequence converges in the dual space of a normed vector space

Suppose that $S=\left\{s_\alpha: \alpha \in A\right\}$ is a set of points in a normed vector space $X$ such that $\overline{span}(S)=X$. If $\left\{f_n\right\}$ is a bounded sequence in $X^*$ and ...
0
votes
1answer
28 views

Fix point of a continuous function under some conditions [closed]

Prove that under each of the following conditions the continuous function $f:[a,b]\to\Bbb{R}$ has a fix point: $f([a,b])\subset [a,b]$ $f([a,b])\supset [a,b]$ When $f$ is bijective and ingective.
1
vote
1answer
21 views

Verify why this is not a metric $d(x,y)=\|x-y\|_p$ ( $\|x\|_p=p^{-h}$ if $x=p^h\dfrac{m}{n}$).

$d(x,y)=\|x-y\|_p$ $p$ is prime and $\|x\|_p=p^{-h}$ if $x=p^h\dfrac{m}{n}$, where $m, n$ are coprimes with $p$. This is not a metric because if $x=y=p^k\dfrac{m}{n}$, then $x-y=0=p^0\dfrac0n$. ...
3
votes
0answers
38 views

The dual of the Annihilator

Let $X$ be a Banach space, and $I$ be a closed subspace. Then it's known that $(X/I)^*=I^{\perp}$. My question is what is the second dual of $X/I$? or what is the dual of $I^{\perp}$ ? If we know ...
0
votes
0answers
24 views

General solution of the recurrence equation with real shifts

How to find a general solution of the following functional (recurrence) equation: $$f(x) = c_1 f(x - a_1) + \dots + c_n f(x - a_n), \tag{1}$$ where $c_i, a_i$, $i = 1, ... n$ are arbitrary real ...
1
vote
0answers
41 views

Distribution function and decreasing rearrangement

Let $(X,dx)$ a measure space and $f\in L^p(X,\mathbb{C})$; let's define its distribution function $$F(\alpha)=meas(\{x\in X||f(x)|>\alpha\})$$ and the decreasing rearrangement ...
1
vote
1answer
27 views

Continuity of an application between function spaces.

I'm trying to prove the following statement... Let $f:[a,b] \times \mathbb{R} \to \mathbb{R}$ a bounded and continuous function, $t_{0} \in [a,b]$, $x_{0} \in \mathbb{R}$, $r>0$ and $$B= \{ x ...
0
votes
0answers
23 views

proof of an relation

Let $\Omega = \mathbb{R}^2_+=\{(x,y)\in \mathbb{R}^2; y>0\}$ et soit $v \in H^1_0(\Omega)$ and let $h \neq 0$. Let $$D_h v = \dfrac{v(x+h,y) - v(x,y)}{h}$$ The questions are: 1- Prouve that ...
0
votes
0answers
45 views

proof some inequalities exercice

Let $\Omega = \mathbb{R}^2_+=\{(x,y)\in \mathbb{R}^2; y>0\}$ and let $v \in H^1_0(\Omega)$ and$h \neq 0$. Let $$D_h v = \dfrac{v(x+h,y) - v(x,y)}{h}$$ The questions are: 1- Prouve that $\forall ...
1
vote
2answers
38 views

Example- $l_p$ norm space

$$||x||= {[{\sum _{i=1}^\infty |x_i|^p}]}^{1/p}$$ Is a norm on $l_p$ space :- space of all sequences made of scalars from $\mathfrak C$(filed of complex numbers). To prove that above is norm on $l_p$ ...
0
votes
0answers
27 views

Approximating a $C^1$ function by piecewise affine maps

Let $\Omega\subset\mathbb R^n$ be an open and bounded domain and let $f\in C^1(\bar\Omega,\mathbb R^m)$. I would like to approximate $f$ by a function $u:\bar\Omega\to\mathbb R^m$ that is piecewise ...
2
votes
1answer
33 views

$f_{n}$ converges in $L^{1}$ and $\|g_{n}\|_{L^{1}(\mathbb R)}\leq \|f_{n}\|_{L^{1}(\mathbb R)}\implies {g_{n}}$ converges in $L^{1}(\mathbb R)$?

Let $f_{n}, g_{n}\in L^{1}(\mathbb R)$ (Lebesgue space). Suppose there exist $f\in L^{1}(\mathbb R)$ such that $\|f_{n}-f\|_{L^{1}(\mathbb R)}\to 0$ as $n\to \infty;$ that is, the sequence $\{f_{n}\}$ ...
0
votes
1answer
45 views

Can I apply Holder's inequality in this case?

I would like to prove that for any $0<q \leq p$, if $x_1 \geq x_2 \geq ... \geq x_n \geq 0$ then $$\left( \sum_{j=m+1}^n x_j^p \right)^{1/p} \leq m^{\frac{1}{p} - \frac{1}{q}}\left( \sum_{j=1}^n ...
0
votes
0answers
53 views

Help with Gronwall's Inequality

I'm trying to solve an extra credit hw problem that has to do with Gronwall's Inequality. I understand (or think I do) how to find an upper bound when $p=0$, but I'm unsure of how to handle the extra ...
1
vote
1answer
27 views

Linear after change of variables

Many interesting non-linear functions of several variables like $\|x\|^2 =\sum\limits_{k=1}^n x_k^2$ become linear in "different variables". Is there a characterization of all (continuous, ...