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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8 views

Prove that the standard orthonormal sequence $(e_n)^\infty_{1}$ is complete in $l^2$.

Prove that the standard orthogonal sequence $(e_n)^{\infty}_{1}$ is complete in $l^{2}$. Where $(e_{n})$ is the sequence with nth component equal to 1 and all others zero.
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6 views

Prove that $\left\|f_n(x)-g_n(x)\right\|^2 = \|f_n(x)\|^2+ \|g_n(x) \|^2-2\operatorname{Re}\int_{\mathbb R} f_n(x)\overline{g_m(x)} dx$

Let $\{f_n(x)\}_{n\in\mathbb Z}$, $\{g_n(x)\}_{n\in\mathbb Z}$ be two sequence of square-integrable functions: $f_n, g_n\in L^2(\mathbb R)$. Prove that $$\left\|f_n(x)-g_n(x)\right\|^2_{L^2(\mathbb ...
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1answer
9 views

Dissipativity for Hilbert spaces

I want to prove that an operator $A:D(A)\to X$ is dissipative $\iff$ $\text{Re}\langle Ax,x\rangle\le 0$ $\forall x\in D(A)$. The proof for this is actually sketched on the Wikipedia page for ...
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1answer
21 views

Is every topological space is measurable?

Actually I am learning about measure theory. But I have confusion between topological space and measurable . Is there any relationship among them or not?
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9 views

prove that a non constant linear functional from a normed linear space X is discontinuous if and only if Z(f) = {x in X | f(x) = 0} is dense in X.

I could prove that if Z(f) is dense in X then the functional must me discontinuous but I am not able to prove the other way round
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1answer
10 views

The w*-extension of a bounded linear functional

Let Y be a Banach space and assume that $X$ is a $w^*$-closed subspace of $Y^*$. Let $f$ be a bounded linear functional on $X$. Does there exist any $w^*$-continuous linear functional $\phi$ on $Y^*$ ...
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0answers
11 views

An example of smooth compactly supported function with non-vanishing Fourier transform

I am trying to find an explicit example of smooth real-valued compactly supported function $u$ in $\mathbb R^n$, $n \geq 2$, such that its Fourier transform $\widehat u$ does not have zeros. By the ...
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0answers
31 views

Fourier Series in Functional analysis

Would you please solve this question? I really have problem with this kind of questions.
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1answer
13 views

Topology on $\mathcal{L}(E,F)$ : infinite dimensional case

We take two infinite dimensional normed space $E,F$, we perform the vector space of continuous linears maps $\mathcal{L}(E,F)$. We use the the sup norm for a linar map. Let $f: U \to F$ where $U$ is ...
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0answers
25 views

How to view Stone-Cech compactification of the real line?

I am going through Arveson's A Short Course on Spectral Theory and have come across an exercise constructing $\beta\mathbb{R}$ using the Gelfand map. I was wondering if there is an explicit ...
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1answer
26 views

Is $W^{1,2}_0$ a Hilbert space?

I came across the term $W^{1,2}_0$. Just a quick question on what the 0 means and is it a Hilbert space? I know $W^{1,2}$ is a Hilbert space. Thanks!
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1answer
36 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 ...
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0answers
13 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 ...
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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 ...
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2answers
53 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
21 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 $ ...
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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 ...
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5answers
59 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 ...
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1answer
64 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 ...
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1answer
36 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 ...
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1answer
22 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, ...
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2answers
49 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 ...
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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 ? ...
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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 ...
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1answer
31 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 ...
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0answers
51 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 ...
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1answer
21 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 ...
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1answer
21 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
18 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 ...
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1answer
38 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 ...
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1answer
20 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 ...
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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 ...
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1answer
21 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 & ...
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1answer
40 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 ...
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4answers
125 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 ...
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1answer
50 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 ...
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0answers
30 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 ...
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0answers
39 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
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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 ...
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2answers
74 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 ...
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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 $$\langle ku,u\rangle = \lambda\langle u,u\rangle$$
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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 ...
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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 ...
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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 $M$. ...
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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!
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0answers
29 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 ...
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0answers
23 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 ...
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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;$E$ ...
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0answers
69 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: ...