Functional analysis is 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, as well as measure, ...

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2
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19 views

On the definition of commuting self adjoint operators.

I'm reading Mathematical Methods in Quantum Mechanics by Gerald Teschl and I came across the following exercise whose statement is causing me some troubles. It goes like this: Let $A$ and $B$ two ...
1
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2answers
30 views

f: R → R and $|f'(x)| ≤ |f(x)|$

Let $f: R → R $ be a function such that $f'(x)$ is continuous and $|f'(x)| ≤ |f(x)|$ for all $x ∈ R$ , if $f(0)=0$ the maximum value of $f(5)$ is My Attempt: I proved that $f'(x)=0$ for $x ∈ [0,1]$ ...
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0answers
11 views

Showing coercivity of the bilinear form associated with a robin boundary value problem

I'm trying to show the existence and uniqueness of weak solutions to the following boundary value problem: \begin{align} -\nabla \cdot ( k \nabla u) &= f \quad \text{in } \Omega \subset ...
1
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1answer
16 views

Find the strong form of a PDE from the weak form.

I'm having a little difficulty understanding how to find the strong form of a PDE given the weak form. For example, I have the weak form as: $\displaystyle\int_\Omega [a(x)\nabla u\cdot\nabla ...
0
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2answers
51 views

Why to introduce norms of vectors?

I am studing Euclidean, metric and normed spaces. What I don't get it is why should I norm a vector. It is usually squared? Why should it be always positive? I've asked this to many people and nobody ...
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0answers
20 views

Show $A=\{x\in \Bbb{R}^n|\sum_{j=1}^{n}|x_j|^p\le 1\}$ is Jordan measurable for $p>0$

Show $A=\{x\in \Bbb{R}^n|\sum_{j=1}^{n}|x_j|^p\le 1\}$ is Jordan measurable if $p>0$. I did show it is a bounded set because if there exists $x^{(N)}\subset A $ such that $||x^{(N)}||\to \infty $ ...
0
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1answer
26 views

If $\|\cdot\|_{1}\le\|\cdot\|_{2}$ then $\|\cdot\|_{2}\le M\|\cdot\|_{1}$ [duplicate]

Let $(X,\|\cdot\|_{1})$ and $(X,\|\cdot\|_{2})$ be complete normed vector spaces and $\|x\|_{1}\le\|x\|_{2}$ $\forall x\in X$. I want to prove that $\exists M>0$ such that $\|x\|_{2}\le ...
4
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5answers
56 views

About inner products, norms and metrics

Do these three kinds of vector spaces, those with an inner-product, those with a norm and those with a metric, are the same sets of vector spaces? At least for finite dimensional vector spaces all of ...
1
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0answers
50 views

Every Hilbert space is isometrically isomorphic with $\ell^2$

Let $H$ be a hilbert space and let $\{u_\alpha\}_{\alpha \in A}$ be a orthornormal basis ($A$ is not supposed to be countable a priori). Then there is an isometric isomorphism between $H$ and ...
1
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1answer
35 views

How to define properly the radial and angular dependence of a function?

Recently I've came across the following situation when studying Quantum Mechanics: suppose we have two operators $A,B$ on the space of functions $L^2(\mathbb{R}^3)$ and suppose they have the following ...
0
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1answer
20 views

How does the general spectral theorem generalize the simpler versions?

I read the following version of the spectral theorem in Banach Algebra Techniques in Operator Theory by Douglas: I'm trying to understand why this is a generalization of the following version, ...
2
votes
2answers
45 views

Why can entire function be written as exponential, and why is it bounded in this way?

Let $A$ be a commutative complex Banach algebra with unit element $e$. Suppose now that $f(x) \in \sigma(x)$ for every $x \in A$ where $\sigma(x)$ denotes the spectrum of $x$. Now, let $x\in A$ and ...
1
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0answers
36 views

The closure of the closed ball is a closed [on hold]

In how many ways you can show that $\overline{\overline{B}(x,r)}=\overline{B}(x,r)$ where $\overline{B}(x,r)=\lbrace y \in \mathbb{R}^n : d_e(x,y) \leq r \rbrace$, and $d_e$ is the euclidean metric ? ...
2
votes
1answer
18 views

Sequential compactness of smooth functions

Suppose I have a sequence $u_n$ of smooth functions on the $N$-dimensional reals. If $\|D^{\alpha}u_n\|_{\infty} \leq C_{\alpha}$ for all multi-indices $\alpha$, then is it possible to deduce that ...
1
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1answer
29 views

Why is $f$ continuous if its kernel is not dense in $A$?

Let $A$ be a commutative complex Banach algebra with unit element $e$. Suppose $X \subset A$ has codimension $1$ and consists out of non-invertible elements. Clearly $X$ is the kernel for some ...
4
votes
0answers
50 views

Is the sum of infinitely many open sets open?

Let $X$ be a locally convex space (or, in particular, a normed space). Let $(O_n)_{n=1}^\infty$ be an infinite sequence of non-empty open sets in $X$ such that the sum $\displaystyle\sum_{n=1}^\infty ...
0
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1answer
20 views

Reference request: Every vector subspace of codimension $1$ of a vector space $A$, is the kernel of some nonzero functional $f$.

I know the following statement is true, but I am looking to find a good reference that proves this quite nicely Every vector subspace of codimension $1$ of a vector space $A$, is the kernel of ...
1
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1answer
20 views

Is this element of $H^1(\Omega)^*$ actually in $L^2(\Omega)$?

Let $\Omega$ be a smooth bounded domain. Let $v \in H^2(\Omega)$ satisfy $-\Delta v = 0$ on $\Omega$ with $\partial_\nu v = g$ where $g \in H^{1/2}(\partial\Omega)$ is normal derivative data. ...
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0answers
12 views

Galerkin method in Sobolev space

I've got this problem to solve: Using Galerkin method, prove that there exists a weak solution of this differential equation: $$-\Delta u = a(x) \circ \triangledown u - u_t +f(x)$$ on $\Omega$ $$u ...
0
votes
1answer
37 views

How to write this down in a rigorous manner?

I'm studying Quantum Mechanics and there's something I'm in doubt in how to write it down in a rigorous way. The idea is: consider a Hilbert space $\mathcal{H}$ and one hermitian operator $A\in ...
0
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0answers
15 views

Compact operators on $L_p$

Consider $M_n(\mathbb{C})$ as $B(\ell_p^n)$ for $n\in\mathbb{N}$ where $p\in[1,\infty)$, and include $M_n(\mathbb{C})$ in $M_{n+1}(\mathbb{C})$ as the upper left corner. Is it true that ...
1
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1answer
8 views

Does continuity in one variable and locally Lipschitz in another imply uniformity in the first?

I understand the definition of Lipschitz functions when talking of functions of single variables. However, I have trouble understanding it when it is a multivariable function. Suppose $ f(t,x):D ...
0
votes
1answer
20 views

Show that $\phi_{y,\epsilon} \in D(\mathbb{R^m})$

For a fixed $y$ $$\phi_{y,\epsilon}(x)= \left\{ \begin{array}{ll} \exp(-\frac{\epsilon^2}{\epsilon^2-|x-y|^2}) & \mbox{if $|x-y| \lt \epsilon$};\\ 0 & ...
0
votes
1answer
38 views

Convergent Bounded Linear Maps

I'm not sure how to show that the composition of two convergent bounded linear maps converges to the composition of their limits. I've shown that the composition of bounded linear maps is a bounded ...
5
votes
4answers
124 views

How do I get $\|x\|\le C\|y\|$ in this case?

I feel that the title is a bit uninformative, please feel free to edit it. This is a problem related to the Open Mapping Theorem. Let $T:X\to Y$ be a bounded linear operator from a Banach space X to ...
1
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1answer
46 views

Fréchet derivative of $f(x) = x$

Im not sure how to find the Fréchet derivative of the function $f : \mathbb{X} \to \mathbb{X}$ given by $f(x) = x$, where $\mathbb{X}$ is a normed space. I'm not given the dimension of the normed ...
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0answers
10 views

$V(A)$ semi group of equivalent projections in $M_∞(A)$ cancelative?

I found in the book of Murphy, C*- Algebras and Operator Theory, the Theorem 7.1.2 : the semi group $V(A)$ of equivalent projections (under Murray Von Neumann equivalence) in $M_∞(A)$ is ...
1
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0answers
28 views

Compact abelian group and Continuous functions

If $G$ is a compact abelian group, $\widehat{G}$ is the dual group of $G$,i.e. all the continuous homomorphism from $G$ to $S^1$,$S^1=\{z\in \mathbb{C}\big | |z|=1\}$. Show that the linear span of ...
0
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0answers
38 views

Let $ (X,\mathcal{A},\mu)$ be a measure space. Prove that $L^\infty (X,\mu) $ is separable.

I tried to prove this but I couldn't. Help me please. Let $ (X,\mathcal{A},\mu)$ be a measure space. Prove that the following statements are equivalent: (i) $L^\infty (X,\mu) $ is separable. (ii) ...
4
votes
1answer
33 views

$\sigma$-algebra generated by weak topology in Hilbert Space

In general, if we have $H$ Hilbert space, and equipped with the weak topology, say $\tau^\ast$, is $\sigma(\tau^*)=\mathcal{B}$?, where $\mathcal{B}$ is the usual Borel $\sigma$-algebra I suspect it ...
1
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2answers
73 views

Why do we need to specify the domain of an unbounded operator?

I am learning about the fluid dynamics and I cam across the following phrase as I was reading about the Stokes operator on Wikipedia. "Since the Stokes operator is unbounded, we must give its domain ...
0
votes
1answer
22 views

Eigen values and self adjoin operator

Can someone give me a clue about how to solve the b part ? All I know is the self adjoint formula $$\langle ku,u\rangle = \lambda\langle u,u\rangle$$
3
votes
1answer
19 views

Showing that the map that takes $u_0$ to solution $u(t)$ is self-adjoint

Let $u$ and $v$ be the solution of the heat equation $$w'(t) - \Delta w(t) =0$$ with initial data $u_0$ and $v_0$ respectively, and with either homogeneous Dirichlet or Neumann BCs on a bounded domain ...
2
votes
1answer
34 views

Eigenvalue equation and separable solutions

Recently, studying Quantum Mechanics I found a doubt regarding separable solutions and eigenvalue equations for differential operators. Suppose we are considering some space $\mathcal{H}$ of functions ...
0
votes
1answer
37 views

A metric between functions on $\mathbb{R}^2$

I want to measure the distance between functions $f$ and $g$ (not necessarily continuous) on a bounded subset $M\subset\mathbb{R}^2$. I assume $f$ and $g$ are locally integrable and bounded on $M$. ...
0
votes
1answer
36 views

What is the name of the following partial differential operator?

What is the name of the following partial differential operator? $$\sum_{|\alpha| \leq n} a_\alpha (\frac{\partial}{\partial x})^\alpha$$ Thank you!
0
votes
0answers
28 views

Does this inequality imply a uniform $L^\infty$ bound?

Suppose I have the estimate for $t > 0$ $$\lVert u(t) \rVert_{L^\infty(\Omega)} \leq Ct^{-1}\lVert u_0 \rVert_{L^1(\Omega)}$$ for the solution $u$ of a parabolic PDE with initial data $u_0$ on a ...
1
vote
0answers
22 views

Galerkin formulation of 1D linear Schrodinger equation

I am interested in the weak (Galerkin) formulation of the 1D Schrodinger equation: $i u_t−\beta u_{xx}=0$ and $u(x,0)=u_0(x)$. As usual, we do integration by parts which yields: $i (u_t,v)+\beta ...
0
votes
1answer
25 views

How to prove $E\|Y'\|\leq E\|Y'-Y''\|,$ where $Y$ is a random matrix in $\mathbb{R}^{n\times n}$, and $Y'$, $Y''$ are independent copies of $Y$?

How to prove $$E\|Y'\|\leq E\|Y'-Y''\|,$$ where $Y$ is a random matrix in $\mathbb{R}^{n\times n}$, and $Y'$, $Y''$ are independent copies of $Y$; $\|\cdot\|$ denotes the $l_2$ operator norm;$E$ ...
0
votes
0answers
68 views

Continuous inclusions in Hilbert-Sobolev space $H^s(\mathbb{R}^n) \hookrightarrow \mathcal{E}^k(\mathbb{R}^n)$

I have to prove that $H^s(\mathbb{R}^n) \hookrightarrow \mathcal{E}^k(\mathbb{R}^n)$ with $s \in \mathbb{R}$, $k \in \mathbb{N}$ and $s-k > n/2$, where $\mathcal{E}^k(\mathbb{R}^n):=\lbrace u: ...
1
vote
2answers
27 views

Constancy of an integral function

Fix some $\ell\in\mathbb{R}^+$. Say that $f:\mathbb{R}^2\to\mathbb{R}_{\geq0}$ and $\mu:\mathbb{R}\to\mathbb{R}^+$ are functions satisfying the following: $f$ and $\mu$ are continuous. $f$ is ...
0
votes
1answer
58 views

Showing that $\displaystyle\underset{n\rightarrow \infty}{\lim}\int_0^1 f_n = \int_0^1\underset{n\rightarrow \infty}{\lim} f_n$

How to solve the following task: Show that if $f_n$ is a sequence of uniformly converging mappings $f_n \in C[0,1]$, where $C[0,1]=\{f:[0,1]\rightarrow\mathbb{R} \;\mid\; f\; ...
1
vote
1answer
49 views

support of an operator on a Hilbert space

Let $x\colon\mathcal{H}\to\mathcal{H}$ be a self-adjoint operator, the support $s(x)$ of $x$ is defined as the smallest projection $e\in B(\mathcal{H})$ such that $ex=xe=x$. Let $x=\int\lambda \, ...
-1
votes
2answers
41 views

Square root and polar decoposition [on hold]

Does every positive definite operator bounded (not necessary compact) has square root? A bounded operator, will AA* has square root(A need to be given compact) Does every bounded operator has polar ...
2
votes
1answer
27 views

Jones construction projections

Let M be a von Neumann algebra with faithful normal normalized trace tr. Let $\{ e_i | i=1,2,\dots \}$ be projections in M such that: $e_ie_{i \pm 1}e_i=\tau e_i $ for some $\tau \leq 1$ ...
2
votes
2answers
95 views

Is $C[0,1]$ reflexive?

I.e. is the embedding $C[0,1]\hookrightarrow \left( C[0,1] \right)^{**}$ surjective? I am having a hard time answering that question. It would be enough to find a closed subspace of $C[0,1]$ which ...
1
vote
1answer
17 views

can two Markov kernels be close in total variation and differ in their ergodicity properties?

This is a question inspired by a recent MCMC paper and linked to an earlier question of Sam Livingstone that did not get any answer. Given two Markov kernels $\mathfrak{K}$ and $\mathfrak{H}$ such ...
1
vote
0answers
18 views

Definition of essential spectrum?

Suppose we have a Hilbert space $\mathscr{H}$ and a bounded linear map $T\in\mathscr{B(H)}$ NOT necessarily self-adjoint. There seems to be loads of definitions of the essential spectrum of $T$. My ...
2
votes
2answers
31 views
+200

Divergence of squared sum of Chebyshev Polynomials $\equiv L+R$ has empty point spectrum

The Chebyshev Polynomials of the second kind $U_n$ are the solutions of the differential equation $$(1-x^2)U_n''(x)-3xU_n'(x)+n(n+2)U_n(x)=0$$ Alternatively they are defined inductively: $$U_0(x)=1 ...
1
vote
2answers
42 views

The set of bounded continuous functions is a closed subspace of that of bounded functions

suppose S is a metric space and $B(S)$ is the set of bounded functions and $C_b(S)$ is the set consisting of bounded continuous functions. Prove that $C_b(S)$ is a closed subspace of $B(S)$. I ...