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Questions tagged [functional-analysis]

Functional analysis, 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 and other topics. For basic questions about functions use more suitable tags like (functions), (functional-equations) or (elementary-set-theory).

2
votes
0answers
28 views

Convergence of a series with terms stemming from a recursive sum of Fourier coefficients of an analytic function.

I'm having trouble seeing if the following series converges: Let $g$ be an analytic $1$-periodic function. Let $g(x) = \sum_k g_k e^{ikx}$ be its Fourier expansion. Recall that the Fourier ...
0
votes
1answer
35 views

Continuity of operators in $L^2$

Let $H=L^2(\mathbb R)$ and $U_tf(x) = f(x-t)$. I want to prove that $t \rightarrow U_tf$ is continuous in $H$ and $t\rightarrow U_t$ is not continuous in $B(H)$. I can't estimate $||U_tf - U_sf||_{L^...
0
votes
1answer
40 views

If $f$ is a Lebesgue integrable function on $[0,1]$ , find polynomials $\{f_n \}_{n=1}^{\infty}$ such that $||f-f_n||_{L^1_{[0,1]}}\to 0$?

If $f\in L^p$ is supported on $[0,1]$ , can we find a sequence of polynomials $\{f_n \}_{n=1}^{\infty}$ such that $||f-f_n||_{L^1_{[0,1]}}\to 0$ ? My attempt: We use $L^p$ denote $L^p_{[0,1]}$ for ...
0
votes
2answers
21 views

Weak Star Convergence of $u_n=\sin(\frac{nx}{n+1})(1+\exp{(-n|y|)})$

Consider the sequence \begin{equation}u_n=\sin(\frac{nx}{n+1})(1+\exp{(-n|y|)}) \text{ where } (x,y)\in I=[-1,1]\times[-1,1], n\in\mathbb{N}.\end{equation} (a) Study the equicontinuity of $(u_n)$ ...
2
votes
1answer
49 views

Corollary of the Hahn Banach theorem

I want to prove the following corollary of the Hahn-Banach theorem. Let $X$ be a normed space. For every closed linear subspace $Y\subseteq X$ and $x\in X-Y$, there exists $x'\in X $ such that $x'|Y=...
4
votes
1answer
52 views

What does the integral of a delta distribution even mean?

Formally, we define $\delta(\phi)=\phi(0)$ where $\phi$ comes from a suitable class of test function. Based on this, the expression $\int_{-\infty}^{\infty} \delta(x) dx$ seems completely meaningless ...
5
votes
1answer
67 views

Is the $p$-norm ever a norm for $0<p<1$?

I wonder: Is there a measure space $(X,\Sigma,\mu)$ such that $L^p(\mu)$ form a normed space w.r.t the $p$-norm, for some $0<p<1$?(assuming that $X$ contains more than point). I know that in ...
3
votes
1answer
47 views

Is a Rational Rotation Algebra a Cutdown of a Matrix Algebra?

Let $\theta=m/n$ and let $A_{\theta}$ be the rational rotation C$^{*}$-algebra with rotation angle $\theta$. I.e., $A_{\theta}=C^{*}(u,v)$, where $u$ and $v$ are unitaries such that $vu=e^{2\pi i \...
0
votes
0answers
20 views

Partial Derivative of Frechet-differentiable Function

Let $X,Y,Z$ be Banach-Spaces and $F:X \times Y \rightarrow Z$ Frechet-differentiable. Then it holds $$ F'(x,y)(u,v) = F_x (x,y)u+ F_y(x,y)v .$$ How do I prove this?
1
vote
2answers
36 views

How calculate extremes of the functional?

Is it also here to use the Euler-Lagrange equation? Could someone tell me how it will look like? $${F}_{u} = \int_{0}^{1} \left( uu' + uu''^{2} + uu'' + u'u'' + 2u'' \right) \mbox{d}x$$ $$u(0) = u'(...
1
vote
2answers
41 views

Vitali's Convergence Theorem but one hypothesis changes

We have the following problem: Let $(Y, \Gamma , \nu)$ be a measure space. Suppose that $\{g_{n}\} \, \subset \, L^{p} \, := \, L^{p}(Y,\Gamma , \nu).$ Prove that $\lim_{n} g_{n} = g$ in $L^p$ if ...
0
votes
2answers
23 views

Is a metrizable and compact subset of $E^*$ is a metrizable complete subset ? for the weak* topology

If we have a Banach space $E$. and we consider it's dual $E^*$, it is a Banach space. so we consider the weak* topology ($\sigma(E^*,E)$) on $E^*$. So My question is : If we have a set $B\subset E^*$,...
1
vote
1answer
39 views

Verapoulous Algebra $C(K) \mathbin{\hat\otimes} C(L)$ is a subalgebra of $C(K\times L)$?

Let $K$ and $L$ be compact spaces. Consider the Banach algebra $V(K,L)=C(K)\mathbin{\hat\otimes} C(L)$ , which is the completion of the $C(K)\otimes C(L)$ with respect to the projective tensor norm. ...
2
votes
3answers
60 views

Proof that the following map $\Phi:\ell^1\to(\ell^\infty)'$ is not surjective

I am working on the dual spaces of sequence spaces, and I want to show that the map $$ \Phi:\ell^1\to(\ell^\infty)',\qquad(\Phi y)(x)=\sum_{i\in\mathbb{N}}y_ix_i $$ is not surjective. I have already ...
0
votes
0answers
19 views

Norm of function of two variables

I know how to find the norm of function of one variable lets say x and obviously it can easily be generalized to higher dimensions. If we have any function $f(x,t)$ than how to calculate norm of such ...
1
vote
0answers
26 views

Existence of the limit of a bounded analytic function

Let $X$ be a Banach space and let $U\subset\mathbb{C} $ be open. If we have an analytic function $f:U\setminus \left\{ 0\right\} \rightarrow X$ such that $\underset{x\in U\setminus \left\{ 0\right\} }{...
0
votes
2answers
64 views

Proving two particular metrics have same topology.

For the metrics $d:\mathbb{R} \times \mathbb{R} \to \mathbb{R}$ defined as $d_{E}(x,y)=|x-y|$ and $$d(x,y)= \frac{|x-y|}{1+|x-y|} . $$ I want to prove this two metrics generate te same topology. By ...
0
votes
0answers
17 views

Prove a given function is an extender form $W^{1,p}(\Omega)$ to $W^{1,p}(\mathbb{R}^2)$

I am trying to prove that given the set $\Omega=\{(x,y) \in \mathbb{R}^2 | y > \sin(x) \}$, the function $E: W^{1,p}(\Omega) \to W^{1,p}(\mathbb{R}^2)$ with $Eu(x,y)=u(x,|y-\sin(x)|+\sin(x))$ is an ...
3
votes
0answers
33 views

$C[0,1], 0<p<1$ is not a norm linear space. What about $p = \infty$ (i.e. supremum norm)? [closed]

To Prove - $C[0,1], 0<p<1$ (with usual p-norm) is not a norm linear space. My thought process - I know that the triangle inequality is violated. But I am not able to produce a counterexample ...
-1
votes
2answers
36 views

The Order of Orthogonality [closed]

I would like to show that $B\subset A$ implies $A^{\bot}\subset B^{\bot}$. Note the meaning behind this: The bigger a subset, the smaller its orthogonal should be. Let $x$ be in the complement of A. ...
-1
votes
0answers
15 views

Non expansive function

Let $F:D \subset X \rightarrow X$ being X a Banach space and $I + \lambda F $ onto for $\lambda >0$ then $R_{\lambda}= (I + \lambda F)^{-1}$ is no expansive. How can i prove this?.
0
votes
0answers
24 views

Practical approach/example for a paper [closed]

My problem is, that I cant say what the different definitions in the paper means in my practical example. My example, and I think its the easiest one, is the process of making a photo, save the ...
5
votes
1answer
56 views

Is the set $\{\delta_x\}_{x \in [a, \ b]}$ a basis for the set of distributions on $C^{\infty}_c([a, \ b])$?

$\def\braket#1#2{\langle#1|#2\rangle}\def\bra#1{\langle#1}\def\ket#1{#1\rangle}$ Is the set $\{\delta_x\}_{x \in [a, b]}$ a basis for the set of distributions on $C^{\infty}_c([a, \ b])$? Below are ...
0
votes
2answers
18 views

Uniform convergence of $f_n$ implies convergence in $\mathcal{L}^1([0,1])$

Let $f_n \in \mathcal{L}^1([0,1])$ for all $n\in \mathbb{N}$ then it follows that $f_n \rightarrow 0$ in $\mathcal{L}^1([0,1])$. The proof goes as follows: Uniform convergence means i): $||f_n-0||_\...
0
votes
0answers
67 views

Closed unit ball and convex hull of its extreme points

Let $K$ denote the closed unit ball of $l_1(\mathbb{N})$ (considered as a vector space over $\mathbb{C}$). (a) Show that the set of extreme points of $K$ is $\mathrm{Ext}(K)= \{ \lambda e_n :\lambda ...
1
vote
1answer
23 views

The image of the weak topology with the canonical injection $J$

Let $E$ be a Banach space. With the weak topology $\sigma(E,E^*)$. And let $J:E\rightarrow E^{**} $ be the canonical injection. Can we prove that the image of the weak topology with $J$ is exactly ...
0
votes
0answers
19 views

how to solve this questions to prove the space is separable.

B[a,b]={f:f is bounded on [a,b]} d(f,g)=sup|f(t)-g(t)|where t belongs to [a,b] then prove that space is not separable.
1
vote
1answer
63 views

Proving $f_n\to f$a.e implies $f_n\to f$ almost uniformly [duplicate]

Exercise: Let $(f)_{n\in\mathbb{N}}$ be a sequence of functions such that $f_n\to f$a.e(almost everyehere) and there exists $g$ integrable such that $|f_n|\leqslant g$a.e for all $n\in\mathbb{N}$. ...
1
vote
1answer
36 views

Limit of series of Heaviside step functions

I couldn't find a better title, because it is a very specific limit that I want to show: $$f_b(y)=\frac{2}{b}\sum\limits_{k=1}^\infty\theta\left(y-\frac{k\pi}{b}\right) \rightarrow f(y)=\frac{2}{\pi}y,...
-1
votes
1answer
26 views

Please Check the convergence of a series [closed]

Let $\{a_n\}$ is converges to z in metric space $(X,d)$. Whether $\sum_{n=1}^{\infty}d(a_n,a_{n+1})$ is converges? If not give an example please
1
vote
0answers
15 views

Laplacian equation on non-compact manifold

Let $(M,g)$ be a complete non-compact manifold with bounded geometry, such that the Sobolev embeddings hold. For the equation $$\Delta u=f,$$ for some $f\in L^2(M)$. Q How can we find a solution $...
2
votes
2answers
37 views

Question about weak topology on normed space

Let $(X,\|\quad\|)$ be a normed vector space, and let $X^\prime$ be the set of all bounded linear maps on $X$. I need help to clarify these questions. Is it correct that the weak topology on $X$ is ...
1
vote
0answers
52 views

Orthogonal projection in Hilbert space

Projector $P\neq0$ is orthogonal projection if and only if satisfied some of the following mutually equivalent conditions: a) $P$ is self-adjoint, $P=P^*$ b) $P$ is normal, i.e. $P^*P=PP^*$ c) $P$ ...
0
votes
1answer
28 views

Application of the open mapping theorem on sequences

Let $X$ and $Y$ be Banach spaces and $A\in B(X,Y)$ surjective operator. I know that from open mapping theorem follow that there exist $C>0$ such that for every $y\in Y$ exist $x\in X$ such that $Ax=...
0
votes
1answer
24 views

When are the topological duals $A^*$ and $B^*$ isomorphic?

If I have to normed vector spaces $A$ and $B$, I was wondering when the topological duals are isomorphic (i.e. $A^* \cong B^*)$ . Is it sufficient that $A \cong B$? Or that $A$ has to be isometric to $...
2
votes
1answer
50 views

If $T_n\to T$ strongly and $x_n\to x$ weakly, must $T_nx_n\to Tx$ weakly?

Let $X$ and $Y$ be Banach spaces, and let $\{T_n\}\subset L(X,Y)$, where $L(X,Y)$ denotes the space of bounded linear operators from X to Y.If $T_n\to T$ strongly and $x_n\to x$ weakly, must $T_nx_n\...
1
vote
1answer
39 views

Show that $(Tu)(x)=\int_{\alpha(x)}^{\beta(x)} u(t)dt$ is Compact linear operator on $C([0,1])$

Show that \begin{equation} (Tu)(x)=\int_{\alpha(x)}^{\beta(x)} u(t)dt \end{equation} is Compact linear operator on $C([0,1],R)$ where $\alpha, \beta:[0,1]\rightarrow [0,1]$ are continuous. My ...
1
vote
2answers
25 views

If $p(x)<1,$ then why does there exists $\alpha \in (0,1)$ such that $\alpha^{-1}x\in M$?

Definition: Given that $E$ is a normed linear space. Let $M\subseteq E$ be an open, convex set with $0\in M.$ For all $x\in E$, define \begin{align} p(x)=\inf\{\alpha>0:\,\alpha^{-1}x\in M \} \end{...
0
votes
1answer
56 views

Is $\left\{ f(x), e^{2\pi i x }f(x) \right\}$ is Linearly independent in $L^2(\mathbb R)$? [on hold]

Let $0\neq f \in L^2(\mathbb R)$ (complex Hilbert space). (1) Can we say that $\left\{ f(x), e^{2\pi i x }f(x) \right\}$ is Linearly independent in $L^2(\mathbb R)$? (2) Is there any known ...
2
votes
0answers
21 views

Obtain a function from local mean values [closed]

(1) Suppose I have a (bounded, measurable) function $f$ on a compact subset $K$ of $R^n$, and for every open interval $U\subset K$ I know only the mean value $$ m_U := \frac{1}{|U|} \int_U f dx. $$ ...
0
votes
1answer
40 views

Weird integral equation with non convolution kernel

Let $f$ and $g$ two rugular functions. My question is the following: Under what condition can we say that for given $g$, there exists $f$ such that we have: $$\int\limits_0^1 {f(x - s,s)ds = g(x)} $$ ...
1
vote
1answer
34 views

$f(x_0)=0 \forall f \in X^*$ ($X^*$ is topological dual) then $x_0=0$.

Let X be a normed vector space of infinite dimension (possibly) and $x_0 \in X$. I would like to show that if $f(x_0)=0 \forall f \in X^*$ (the topological dual) then $x_0=0$. I asked this question ...
0
votes
1answer
36 views

About Riesz representation theorem

Let $X=l_p^{(3)}$, where $1\lt p \lt \infty$, and $\phi(x) = x_1-2x_2+3x_3$. Decide whether $\phi$ is bounded, and if so, find $||\phi||$. So by marking $y=(1,-2,3)$, we can see that $\phi(x)=\...
5
votes
1answer
56 views

If $X$ is an Ito process, is $\mathbb E(\int X \mathrm d X)$ convex?

Consider the functional $F$, which is defined for each Ito process $$X(t) = \int_0^t \mu(s) \mathrm d s + \int_0^t \sigma(s) \mathrm d W(s)$$ as $$F(X) := \mathbb E\bigg(\int_0^T X(s) \mathrm dX(s)\...
1
vote
1answer
52 views

Properties of laplace type transform of $t^{\alpha - 1}$

Let $p>2, \frac{1}{p} < \alpha < 1- \frac{1}{p}$ and define $g_\alpha(t) := t^{\alpha - 1} \chi_{[1, \infty]}$. Then $g_\alpha \in L^p(\mathbb R)$. Define $$f(z) := \int_1^\infty g_\alpha(t) \...
1
vote
3answers
41 views

$f(x_0)=0 \forall f \in X^*$ then $x_0=0$.

Let X be a vector space of infinite dimension (possibly) and $x_0 \in X$. I would like to show that if $f(x_0)=0 \forall f \in X^*$ then $x_0=0$. So the first intuition would be to pick a function ...
1
vote
1answer
52 views

Uniform convergence for operator of translation

For $a\in R^d$, let $T_af(x)=f(x-a),$ for all $f\in L^p(R^d), 1\leq p<\infty$ and all $x\in R^d$. I need example that this operator doesn't converge uniformly when $a\rightarrow 0$. I know that ...
1
vote
0answers
54 views

Riesz lemma for $L^p$ space

I need a proof for special case of Riesz lemma (when $\varepsilon$ is 0): If Y is a closed proper subspace of $L^p(\mu)$ for some $1<p<\infty$, then there exist $f\in L^p(\mu)$ such that $||f||...
1
vote
0answers
51 views
+50

using zeta function in the regularization of functional trace and determinant

I'm looking for books or lectures concerning Zeta function regularization. In particular, I'm interested in using zeta function in the regularization of functional trace and determinant . To be more ...
0
votes
1answer
24 views

Prove convolution of (f - Lipschitz, bounded) and some (g $\in L^1$) fulfills the Lipschitz criterion [closed]

Prove that the convolution of bounded, Lipschitz-continuous function f : $\mathbb R^n \rightarrow \mathbb R$ and function $g \in L^1(\mathbb R^n)$ fulfills the Lipschitz criterion itself.