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).
49,236
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Relationship between the Fundamental Lemma of Calculus of Variations and Completeness in Statistical Inference
I've been studying the Fundamental Lemma of Calculus of Variations and the concept of completeness in statistical inference, and I've noticed that both concepts involve the idea that an integral (or ...
- 39
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I have a question about finite dimension norm linear spaces
Let R is linear space over the field Q and Q is a finite dimension subspace of R the what about the completeness of Q ?? Since we know that any finite dimension norm space is complete but here Q is ...
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Why dual norm inequality is tight in $\mathbb{R}^{n}$
Let $\lVert \cdot \rVert $ be any norm on $\mathbb{R}^n$
I want to prove that
For any $x$, there is some $z$ such that $z^{T}x = \lVert x \rVert \lVert z \rVert _{*}$
where $\lVert y \rVert _{*} = \...
- 119
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2
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Is there a 'simple' function that flips the order of positive numbers without making them negative?
If I want to flip the order of some numbers, I can just multiply them with -1. But is there a not too complicated way to do it such that the numbers remain positive?
Here's my attempt to word the ...
- 320
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Is $f\in L^1$ approximated (in the sense of $L^1$) by a sequence of continuous functions?
For arbitrary $f\in L^1$, is there exist a sequence of continuous functions $\{f_n\}_{n=1}^\infty$ s.t. $\lim_n \|f-f_n\|_1=0$ ?
I think this statement is true.
Let $f\in L^1.$
Since $C_c$ is dense in ...
- 411
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Different measurability of Hilbert-space valued random variable
My question is motivated by this link.
Let $(\Omega,\mathcal{F})$ and $(Y,\mathcal{B})$ be measurable spaces, a measurable map $T:\Omega\to Y$ is called a $Y$-valued random variable.
Now let $H$ be a ...
- 12.7k
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Mean minimizes squared error in normed vector space
The goal is to show the cost function $$J(x) = \sum_{k=1}^n ||x - x_k ||^2$$
is minimized when $x = m$, where $m$ is the sample mean $m = \frac1n \sum_{k=1}^n x_k$
I would like to stay as general as ...
- 23
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1
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On the convergence of approximate units for C*-algebras.
Let $A$ be a non-unital C*-algebra and let $\pi : A \to \mathcal{B}(H)$ be a non-degenerate representation of $A$ (that is, $\mathrm{ span }\{\pi(a)h : a \in A, h \in H\}$ is a dense subset of $H$). ...
- 853
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Rudin's RCA Fourier Coefficients of $L^1$ - functions
we associate to every $f$ $\in$ $L^1(T)$ a function $\hat f$ on $Z$ defined by
$\hat f $ $=$ $\frac {1}{2\pi}$ $\int_{-\pi}^{\pi}$ $f(t)$$e^{-int} dt$ $(n \in Z)$.
It is easy to prove that $\hat f $ $...
- 751
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Pointwise estimates for convolution with heat kernel
If $$h_t(x) = (2\pi t)^{-1/2}\exp(-x^2/(2t))$$ is the heat kernel in dimension $d = 1$, something I noticed is that if I fix a time, say $t = 1$, and take a function $f$, say $f(x) = x^2$, then $$P_1f(...
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Why does a multiplicative functional $m \in \Delta_A$ extend uniquely to $m^e \in \Delta_{A^e}$?
The setup is as follows. $A$ is a commutative Banach algebra, and $\Delta_A$ is the structure space of $A$, consisting of all non-zero continuous algebra homomorphisms $f:A\to \Bbb C$. The text says ...
- 11.4k
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Trying to prove directly that the Laplacian is closed.
I am trying to prove directly that the Laplacian operator is closed on its maximal set. I have the following with some doubts.
Considering the operator $-\Delta$ defined on $D(-\Delta)=\left\{u\in L^2:...
- 2,876
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1
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Proving or disproving a space is a Banach Space
Here's my question:
Suppose $(X , \lVert \cdot \rVert)$ is a Banach Space, $M$ be a
subspace of $X$. Assume that $N$ is closed subspace of $X$ such that
$M + N = X$ and
$M \cap N = \{ 0 \}$ Then ...
- 3,605
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Proof of $A\psi = \lambda \psi \Rightarrow h(A)\psi = h(\lambda)\psi$ on Borel functional calculus
Let $A: D(A) \to \mathscr{H}$ be a densely defined unbounded self-adjoint operator on a separable Hilbert space $\mathscr{H}$. By the Spectral Theorem, there exists some finite measure space $(M,\mu)$,...
- 1,601
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Convergence in the formulation of the spectral theorem
Let $\mathcal H$ be a complex (separable, if needed) Hilbert space and $B(\mathcal H)$ the ring of bounded operators on $\mathcal H$. I am interested in understanding the formulation of the spectral ...
- 6,554
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40
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Is the space of bounded monotonic continuous functions on $\mathbb{R}$ separable?
Let $C_b(\mathbb{R})$ be the space of bounded continuous functions on $\mathbb{R}$ to $\mathbb{R}$. It is known that the Banach space $X = (C_b(\mathbb{R}), \Vert \cdot \Vert_\infty)$ is inseparable.
...
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1
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What are the $\mathcal{M}_{sym}=\{E \operatorname{Lebesgue measurable and } E=-E \}$ measurable functions?
I am supposed to consider $L^2(-1,1)$ and the subspace $V= \{ u \in H : u \operatorname{is} \mathcal{M}_{sym}-measurable \}$ where $\mathcal{M}_{sym}$ is the $\sigma-$algebra generated by: $\{E \...
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Duality of $H^1$ and BMO.
While proving that the dual of $H^1$ is $BMO$ in Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, page 143, Stein says that we have $\left\Vert g \right\Vert_{H^1} \...
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If $\varphi$ is a normal faithful semifinite weight, is $\eta_\varphi(\mathfrak{n}_\varphi\cap\mathfrak{n}_\varphi^*)$ dense in $\mathfrak{H}_\varphi$
Let $M$ be a von Neumann algebra and $\varphi: M_+\to [0, \infty]$ be a normal, faithful semifinite weight. Consider its associated semi-cyclic representation
$$\pi_\varphi: M\to B(\mathfrak{H}_\...
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Continuity of the fractional Laplacian operator
Considering $(-\Delta)^s: H^s(\Omega) \to L^2(\Omega)$, is it possible to show that this operator is closed (continuous)?
For instance, taking a sequence $(u_n) \subset H^s(\Omega)$ with $u_n \to u$ ...
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2
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27
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How can I calculate the dual norm in this case?
We define $E$ the space of continuous functions as $x: [0,1] \rightarrow \mathbb{R}$, endowed with the norm for $x \in E$:
$$\|x\|_{\infty} = \max_{t\in [0,1]}|x(t)|$$
It is further defined $T : E\...
- 3
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Question on showing that $\frac{1}{n}\sum_{k=0}^{n-1}U^kf$ converges in norm to the orthogonal projection $Pf$ to the space $\{f\in H: Uf = f\}$
Edit: The reference I am reading is Yves Coudène's Ergodic Theory and Dynamical Systems, chapter 1, proof of theorem 1.1 on page 5.
Let $H$ be a Hilbert space and $U:H\to H$ a bounded linear operator ...
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37
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Proof of $\limsup_{n\to\infty} \|a^n\|^{1/n} \le r(a)$
Theorem $2.2.6$ in Principles of Harmonic Analysis by Anton Deitmar and Siegfried Echterhoff proves the well-known formula for the spectral radius in a unital Banach algebra, namely $r(a) = \lim_{n\to\...
- 11.4k
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Calculate the operator norm defined with the dependency on the function f(t)
Let $E=F=G=C^0([0,1])$ all three endowed with the norm $||\cdot||_\infty$ of the uniform convergence defined as $||x||=\max_{t\in[0,1]}|x(t)|$.
Let $f\in C^0([0,1])$ fixed with $f(t)\geq0 \ \forall t\...
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A product of a $H^1_0(\Omega)$ function with a function defined on $S^{N-1}$ is also in $H^1_0(\Omega)$.
Let $\Omega = B_1(0)$ the ball centered at $0$ with radius equal 1 and $u \in H^1_0(\Omega).$ Also consider $f \in C^\infty(S^{N-1})$. I'm trying to prove the function $g(x) := u(x) f(x/|x|)$ is in $H^...
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Show that $f_n(x) = \frac{\sqrt{n}x}{1+nx+n^2 x^3}$ does not converge in C([0,1]) with the $\sup$ norm
Show that $\displaystyle f_n(x) = \frac{\sqrt{n}x}{1+nx+n^2 x^3}$ does not converge in C([0,1]) with the $\sup$ norm.
I first find the pointwise limit. We have $f_n(0)=0$, and for $0<x\le 1$, we ...
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Monotone Class Theorem and Stone-Weierstrass
We're talking about th eMonotone Class Theorem in my probability course and I've noticed some similarities with the Stone-Weierstrass Theorem. I'm told there's a number of different versions of the ...
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Lipschitz continuity of translation in $L^p$
Let $f\in L^p(\mathbb{R})$ for $1\leq p <\infty,$ then $||f(x+h)-f(x)||_{L^p(\mathbb{R})} \rightarrow 0$ as $h\rightarrow 0$.
Furthermore, for $p=1,$ if $u\in L^1(\mathbb{R}) \cap W^{1,1}(\mathbb{R}...
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1
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singular value decomposition of sum
Let $A,B$ be positive, linear trace class operators on some Hilbert space.
I would like to know if the following trace inequality for some $\mu>0$ is true
$$
\mathrm{Tr}\!\left(A\,(A+B+\mu I)^{-1}\...
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2
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1
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26
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Does the shadowing property hold for the rotation?
Let $\alpha\in \Bbb{R}$ and consider the rotation $R_\alpha: \Bbb{R}/\Bbb{Z}\rightarrow \Bbb{R}/\Bbb{Z}$ a.t. $x\mapsto (x+\alpha)\operatorname{mod}\Bbb{Z}$. I want to check if $R_\alpha$ satisfies ...
- 1,772
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1
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How to bound the integrand to show that $|f_u(x,u,\nabla u)|\le 1+ |u|^{p-1}+|\nabla u|^{p-1},$ implies the E-L term $f_u(x,u,\nabla u) \in L^{p'}$
From the book "introduction to the calculus of variations", DACOROGNA
I am having trouble giustifying how the last inequality below comes out. p and p' are conjugate exponents
Given the ...
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1
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$B(0,1) $ isn't totally bounded in $\ell^{1}$ [closed]
I've been able to show using Riesz's lemma that $B[0,1]$ isn't totally bounded , and I tried to solve it for $B(0,1)$ , but I wasn't capable of solving it.
Any suggestions?
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References a ideas behind weight functions [closed]
I have some questions about weight functions:
if we consider the $L^2$ norm with a weight of a function $f$:
$$\int_\mathbb{R} w(x)|f(x)|^2 dx,$$
what is the idea of considering the weight function?
...
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What are all $L^pL^q$ estimates for the heat equation (with gain of derivatives)?
The heat equation and the heat kernel.
Consider the heat equation on $\mathbb R$:
$$ \left\{\begin{aligned}u_t-\Delta u&=f\\u(0,x)&=0\quad\forall x\in\mathbb R. \end{aligned}\right. $$
It is ...
- 2,523
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Norm of Dirac delta derivative in negative order Sobolev space
Let $k\in \mathbb{N}$, and consider the negative order Sobolev space $W^{-k,p}$. Given $p>1$, a distribution $f$ is in $W^{-k,p}$ if and only if, by definition of the space,
$$
f=\sum_{|\alpha|\leq ...
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Functions that can be approximated by derivatives of test functions
Let $I \subseteq \mathbb{R}$ be a compact interval. We know that functions in $L^p(I)$, $(p \geq 1)$ can be $L^1$-approximated by a sequence $(\varphi_n)_{n \in \mathbb{N}}\subseteq C_0^\infty(I)$ (...
- 5,748
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Showing that this hermitian functional is positive [duplicate]
Let $\rho$ be a hermitian functional on $C(X)$ where $X$ is a compact and Hausdorff space. Suppose $\|\rho\|=\rho(1)$. Then show that $\rho$ is a positive linear functional on $C(X)$. Part of the ...
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Let $T,S$ be unilateral shifts in $H,K$ and $A\in B(H,K)$ a contraction. If $S^*A=AT^*$, then why is $A$ a transposed infinite Toeplitz matrix?
Let $H, K$ be Hilbert spaces.
As the Toeplitz Matrix, I define an operator $P_n$ in the form:
$$P_n = \begin{pmatrix}
Q_0 & 0 & 0 & \ldots & 0 \\
Q_1 & Q_0 & 0 ...
- 513
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0
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8
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$A(f)$ has a minimum in the rectangle $R_{\rho, \gamma}=\left\{f: f \in \mathscr{H}, J(f) \leq \rho,\left\|f-f_0\right\|_0 \leq \gamma\right\}$
I'm working on a proof of the existence of a minimum of a linear form in a $\left.C_p=\{f: f \in \mathscr{H}, J(f) \leq \rho\}, \forall \rho \in\right] 0, \infty[$.
One step of the proof is that we ...
- 217
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1
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31
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Rudin’s RCA A Convergence Problem Fourier Series
There is it: Is it true for every $f\in C(T)$ that the Fourier series of $f$ converges to $f(x)$ at every point $x$?
Let us recall that the $n$th partial sum of the Fourier series of $f$ at the point $...
- 751
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0
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33
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Show that a sum of indicator function is continuous
let $\epsilon_1, \epsilon_2, ... \epsilon_n \in (0,1)$ such that $\epsilon_1 <\epsilon_2 < ... <\epsilon_n$
how can I show rigorously that :
-the function $f(x) = \sum_{k=1}^{n} \mathbb{1}
_{\...
- 1
4
votes
1
answer
60
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Weak derivative of a step function
Consider the function
\begin{align} v(x)=\begin{cases}1&~\text{ if }x\in (0,1)\\ k&~\text{ if }x=0\\ -1&~\text{ if }x\in (-1,0)\\0&~\text{ if } x\geq1 \text{ or }x\le-1\end{cases}\end{...
- 83
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0
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61
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Fourier Series Expansion and its derivatives
I am studying the Fourier series in my real analysis course and I am stuck on some questions...
I have a function $f(x)$ in the trigonometric system given as:
$f(x)\sim \frac{a_0}2+\sum_{n=1}^\infty\...
- 97
1
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2
answers
63
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Show that $P$ is normal if and only if it is an orthogonal projection.
Let $V$ be a finite dimensional complex inner product space, and $P$ be a projection. Show that
$P$ is normal if and only if it is an orthogonal projection.
My work:
For the statement "$P$ is ...
- 934
0
votes
1
answer
61
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If $T$ is a positive operator, and with $S^2=T$, then $S$ must be self-adjoint.
Let $S$ and $T$ be a linear operator on finite dimensional vector space $V$. If $T$ is a positive operator, and with $S^2=T$, then $S$ must be self-adjoint. Does this statement hold?
I try to use the ...
- 934
2
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0
answers
44
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Show that finite dimensional topological subspaces are topologically isomorphic to $\mathbb{C}^n$ using induction.
Let $X$ be a topological Vector space and $Y$ a finite dimensional subspace with $\text{dim}Y = n$, $n \in \mathbb{N}$. Now, one can show that every Isomorphism between $\mathbb{C}^n$ and $Y$ is a ...
- 383
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0
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19
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Annihilator of pre-annihilator and pre-annihilator of annihilator in Banach space
Let $X$ be Banach space. Prove that $(Z^{\perp})^{\perp\perp} = \overline{Lin Z}$ and $\overline{Lin Y} \subset (Y^{\perp\perp})^{\perp}$, where $Z^{\perp}$ is annihilator of $Z \subset X$, and $Y^{\...
- 161
3
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2
answers
59
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Weak convergence of functionals $g_n^*(f) = n\int_0^1 x^nf(x)dx$ [closed]
Show that sequnce of functionals $g_n^*(f) = n\displaystyle{\int_0^1 x^nf(x)dx}, f \in C[0,1]$ converges weakly and find its limit functional. Does it converge in the norm of space $C^*[0,1]$?
I don'...
- 161
1
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1
answer
35
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Weak, strong and uniform convergence of operator on $L^p(\mathbb{R}^d)$ [duplicate]
For every $a \in \mathbb{R}^d$, let $T_a(f)(x) = f(x-a), \forall f \in L^p(\mathbb{R}^d), \forall x \in \mathbb{R}^d$. Prove that $T_a$ is a linear isometry of space $L^p$ on itself. Find $\lim_{a\to ...
- 161
4
votes
1
answer
53
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Variational formulation, weak formulation
I'd like to find the weak formulation of the problem
$-u''+au=f$ on $(0,1)$
$u(0)=0$
$u'(1)=b$
$a>0$
and show that there exists a unique solution using Lax-Milgram.
What I did:
By multiplying ...
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