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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Pointwise limit of convolution

Suppose $\omega$ is the standard mollifier in $\mathbb R$. Then, let $\omega_{\epsilon} (x):= \frac{1}{\epsilon} \omega \left(\frac{x}{\epsilon}\right)$. For $0 < t_{1} < t_{2}$ the following ...
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1answer
19 views

The weak topology on an infinite dimensional linear space is not first-countable

I thought I needed help proving the above statement, but during typing I found a proof. Since I had already written it all down I will post it anyway, maybe in the future someone can benefit from it. ...
0
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1answer
21 views

Is there any inner product on $M_{n \times n}$ inducing this norm?

The set $M_{n \times n}$ is the collection of all $n \times n$ matrices over $\mathbb{R}$. Definition: $\|A\|_2=Sup_{\|u\|_2=1} \|Au\|_2$. Is there any inner product on $M_{n \times n}$ inducing ...
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1answer
18 views

Verification of some conditions and facts of the Laplacian on a Riemannian manifold

I came across the following introduction to a paper I was reading: "Let $L$ be a second-order, elliptic, differential operator, with smooth coefficients, on a Riemannian manifold $M$. Assume $-L$ is ...
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2answers
130 views

How to apply the Gronwall lemma

Consider $x'=f(x)$ such that $(x_1,x_2)\mapsto(-x_1+2x_2,-2x_1-x_2)$. Show that for two solutions $x(t)$ and $y(t)$ of the above differential equation, we have: $$\lVert x(t)-y(t)\rVert \leq ...
4
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49 views

How numerical radius help us to conclude an operator is normal and partial isometry?

In Furuta's book, "Invitation to Linear Operators" there is a theorem, theorem 2 in 3.7.3, that says: If $T^k=T$ for some integer $k\ge 2$ and if $w(T)\le 1$, then $T$ is the direct sum of a unitary ...
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5 views

Embedding the Schreier spaces into $C(\alpha)$ for some countable ordinal $\alpha$

Let $1\leq\xi<\omega_1$ be any countable ordinal, and denote by $\mathcal{S}_\xi$ the Schreier family of order $\xi$. Then the Schreier space of order $\xi$ is the completion $S_\xi$ of $c_{00}$ ...
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28 views

Frechet mean of $k$ elements in the n-dimentional sphere.

The Frechet mean of $k$ elements $x_1, \ldots, x_k \in S = \{ x \in \mathbb{R}^n,\, \| x \| = 1 \}$ is defined as the $\text{arg}\underset{\|x\|=1}{\text{min}} \sum_{i=1}^n d^2(x_i,x)$. Where ...
4
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1answer
115 views

Soft question: what are some elementary motivations of using functional analysis to study probability theory?

Recently I've become curious about the links between functional analysis and probability theory. What are some simple reasons why a functional analytic approach is preferable to a measure-theoretic ...
7
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291 views

References on the Nash-Moser implicit function theorem

To learn, the Nash-Moser implicit function theorem, I tried the document Hamilton (1982) The Inverse Function Theorem of Nash and Moser, but the article is very encyclopedic. I have a ...
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1answer
65 views

Implications of inner products vs normed spaces vs metric spaces

Is it true that: -an inner product satisfies the properties of a norm if and only if the norm satisfies the parallelogram equality -a norm can be induced by a metric if and only if the metric ...
4
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1answer
316 views

Properties of the square norm in Banach spaces

Let $X$ be a Banach space with its dual $X^*$. Consider the mapping $f: X\rightarrow \mathbb{R}$ given by $$ f(x)=\frac{1}{2}\|x\|^2. $$ We have know that when $X$ is a real Hilbert space ($X=X^*$) ...
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21 views

The highest direction of the trace operator

Let $W$ be a real and symmetric matrix ${m \times m}$ from the set $\widetilde{W_m}$, and $T:\widetilde{W_m} \rightarrow \mathbb{R}$ a function defined by $T(W) = trace(W^3)$. We are interested to ...
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1answer
36 views

Vector Space that is a Sequence Space

Consider the sequence $(x_n)_{n \in \Bbb N}$ $ |$ $ x_n=\sum_{i=0}^{n}f(i)$ then if $f(i)$ can also be viewed as a sequence $(y_n)_{n \in \Bbb N}$ $ |$ $y_n=f(n)$. Both of these sequences ...
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2answers
48 views

Analytic vectors of self-adjoint unbounded operators

I am working with self-adjoint unbounded linear operators on infinite-dimensional separable Hilbert spaces, and I would like to know some references regarding analytic vectors, especially with respect ...
2
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3answers
37 views

Can distributions be thought of as functions of a real variable?

I understand that, given some function space, distributions lie in the dual space. In that sense, they can be thought of as functions of a "function of a real variable" variable. But the common ...
2
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1answer
48 views

Correctness of proof that weak convergence implies pointwise convergence in C([0,1])

I want to prove that in the space of (complex-valued) continuous functions on the real interval [0,1] equipped with the sup norm, which I will denote by $\mathscr{C}([0,1])$, weak convergence implies ...
0
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1answer
32 views

Euclidean geometry and $L_2(\lambda)$ space

Suppose $f,g\in L_2(I,\lambda)$ with $\lambda$ any probability measure and the norm $\| x\|=\sqrt{\langle x, x\rangle}$. Could we have the same geometric properties in this space as in the Euclidean ...
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0answers
31 views

dimension set of all fourier series functions and basis

Let $F$ be the set of all functions on $D = [0,2 \pi] \times [0,2 \pi]$ that have a convergent Fourier series. Is the dimension of $F$ equal to $\aleph_{0}$? Given a countable infinity of linearly ...
2
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1answer
30 views

Do approximate identities remain approximate identities if one adjoins 1 to a C* Algebra?

If we have a C* Algebra $\mathscr{U}$ without an identity we can adjoin an identity $\mathbb{1}$ in the following way: We take $\mathscr{\tilde U}$ to be the set $\{(\alpha,A); \alpha \in ...
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1answer
20 views

Poisson functional on bounded domain

I was wondering if it is actually clear that on bounded domains the Poisson integral is bounded from below: $$I[u]=\int_{\Omega} \left( \frac{1}{2}\lvert \nabla u \rvert^2 - u\rho \right)\, dx,$$ I ...
3
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0answers
39 views

Find a function which satisfies an integral equation

How can I find a function, $$ f: \mathbb{R} \to \mathbb{R} $$ which satisfies the following equation: $$\cos\left(t^2\right) = \int_{-\infty}^{\infty} e^{itx}f(x)\,dx$$
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59 views

Linear Independence for functions defined by integration [on hold]

Given that the set of functions $$f_i(x,y), \quad i=1,\dots,n$$ are linearly independent for $(x,y) \in [0,1]^2$. Is the set of functions, $g_1,\ldots,g_n$, defined by $$ g_i(x) = \int_{y\in [0,1] } ...
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1answer
13 views

An inequality for power of positive functions

Let $f,g,h$ be positive real vlaued functions on a finite set $\mathbb{X}$. Let $p >1$. I am wondering whether the following should be true? $$\sum_{x\in ...
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2answers
23 views

Understanding Quotient spaces - Shrinking down

I am looking at Page 57 of Kreyszig's Functional Analysis, and I have been given an exercise: Let $X=\Bbb R^3$ and $Y=\{(\xi_1,0,0)| \xi_1 \in \Bbb R\}$ 1) Find $X/Y$: So $X/Y=\{[x]:x+Y,\forall ...
3
votes
1answer
64 views

Prove a sequentially compact metric space is bounded.

Prove that if the metric space $(X, d)$ is sequentially compact, that there exists points $x_0$ and $y_0$ belonging to $X$ such that; $$d(x, y) \leq d(x_0, y_0)$$ for every $x$ and $y$ belonging to ...
0
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0answers
38 views

Limiting value of $L^2$ functions

Let $f\in L^2(\Omega)$, where $\Omega \subset \mathbb{R}^2$ is the unit square $[0,1] \times [0,1]$. Let $x\in \Omega$. Suppose I evaluate $f$ at points from some direction that approach $x$. ...
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2answers
450 views

Can a dynamic programming problem be transformed into a linear algebra problem?

Here is a simple standard economic problem: Let Robin Crusoe have an endowment $w_0$ of lembas bread. She is immortal and discounts at a rate of $\beta$ per period. Each period (from $t=0$ on) she ...
2
votes
1answer
18 views

The convergence of a product of sequences converging in $L^2$.

Earlier today I found myself pondering the following question for which I do not have a reasonable answer. Suppose $f_m\to f$ and $g_m\to g$ in $L^2$. Moreover suppose that $f_m g_m\in L^2$ for ...
5
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1answer
636 views

Strict Inequality in Rudin's Proof of the Riesz Representation Theorem

In Rudin's proof of the Riesz Representation Theorem (step ten), he proves that $$\Lambda h_i \leq \mu(V_i) < \mu(E_i) + \epsilon/n , \quad \mu(K) \leq \sum\limits_{1 \leq i \leq n} \Lambda h_i.$$ ...
1
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1answer
66 views

Given an arbitrary sequence {$x_n$} in $\Bbb{R}$, find a test function $f$ with $f^{(n)}(0)=x_n$

Given an arbitrary sequence {$x_n$} in $\Bbb{R}$, can I find a test function having the $n$-derivative equal to $x_n$ at $0$?
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14 views

The tensor product of $ {L^{1}}(G) $ and a Banach space

Let $ G $ be a locally compact group and $ A $ a Banach space. It is known that the tensor product $ {L^{1}}(G) \otimes A $ is isometrically isomorphic to $ {L^{1}}(G,A) $. I need a proof of it.
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1answer
27 views

Radius of convergence of Taylor expansion of $z \mapsto (1 - z \cdot a)^{-1}$

Let $A$ be a Banach $\mathbb{C}$-algebra with norm $\text{N}(-)$ and let $a \in A$. Where can I find a reference to/can somebody supply a proof of the following posited equality?$$\max_{z \in ...
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0answers
34 views

Is there anything called kernel space?

Here I am referring kernel as an integral operation.The wikipedia link is this https://en.wikipedia.org/wiki/Integral_transform My question is: consider the function insider the integral $f(t)$ is ...
3
votes
1answer
51 views

A simple $C_{0}$-semigroup question.

Let $u:[0,t_{e}]\to\mathcal{D}(A)$ satisfy $$\begin{cases} \frac{du}{dt}=Au & 0\le t \le t_{e} \\ u(0)=x \end{cases}$$ I want to prove that necessarily $u(t)=T(t)x$. So it's clear to see that ...
3
votes
1answer
52 views

Elegant way to prove that the space must be infinite dimensional?

Let $F(S,V)$ be the set of all functions from S to a vector space V, assume that $V\ne\{0\}$, and that S contains infinitely many elements, then we must have that $F(S,V)$ is ...
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2answers
50 views

$f(I)\cap g(J)\not=\phi$ for all open interval $I,J$

Let $f,g:\mathbb{R}\rightarrow \mathbb{R}$ and $f(I)\cap g(J)\not=\phi$ for all nonempty open interval $I,J$. Consider $f_1=\chi_\mathbb{Q}$ and $g_1=\chi_\mathbb{Q^c}$, we know that $f_1$ and $g_1$ ...
1
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1answer
34 views

Negativity of Convex Combinations

Consider the functions $f(x)$, $g_1(x)$ and $g_2(x)$ with following properties: $\int f(x) dx =\int g_1(x)dx =\int g_2(x)dx =1$. Define the following measure of negativity for the functions: ...
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votes
1answer
46 views

How can I prove that $(X,τ)$ is a Hausdorff topological space?

Let $(X_1,τ_1)$ is a Hausdorff topological space and $(X_2,τ_2)$ is a Hausdorff topological space and $X=X_1\times X_2$ and $τ$ The product topology How can I prove that $(X,τ)$ is a Hausdorff ...
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0answers
22 views

The restriction of a discontinuous linear functional to any open set is surjective.

Problem. Let $X$ be a topological vector space and $f:X\to\mathbb{K}$ a linear mapping. Prove that if $f$ is discontinuous, then $f(A)=\mathbb{K}$ for all nonempty open set $A\subset X$. I'd like ...
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1answer
23 views

Is probability mass function (PMF) the “law of X”?

Are they two the same? If not, what's the differences between these two? In continuous case, is PMF also equal to the integration of probability density function?
2
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1answer
44 views

Show that the spectrum of an operator on $\ell^2(\mathbb{N})$ is $\{0\}$.

The problem I have comes from Walter Rudin's Functional Analysis, chapter 10 exercise 19. The exercise begins with the following: Let $S_R$ be the right shift operator, acting on ...
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0answers
56 views

Specific Root of Interpolating Polynomial

We define polynomial $P=(x-\beta)\cdot g(x)$, where degree of $P$ is fixed $n-1$, $\beta$ is chosen uniformly at random from the field of $p$ elements. We evaluate $P$ at some $x_i$ values. So we get ...
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10
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Equivalent Definitions of the Operator Norm

Would you give me a proof of the equivalence of these ones? $$\begin{align*} \lVert A\rVert_{\mathrm{op}} &= \inf\{ c\;\colon\; \lVert Av\rVert\leq c\lVert v\rVert \text{ for all }v\in V\}\\ ...
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0answers
27 views

Given a spectrum, what can we know about its function?

Say we are given a well-behaved function $f(t)$ (either discrete or continuous) and are able to compute its spectrum $F(k)$ using the (discrete) Fourier transform. Then say we lose $f(t)$ and know ...
4
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2answers
60 views

How to simplify $ \int_{\Bbb{R}^2}\Delta\varphi(x)\log|x|^2\ dx $ using Green's indentity?

Let $\varphi\in C_c^\infty(\Bbb{R^2})$ (infinitely differentiable functions with compact support) and consider $$ I=\int_{\Bbb{R}^2}\Delta\varphi(x)\log|x|^2\ dx, $$ the existence of which is ...
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0answers
15 views

The dual of the space of $p$-locally integrable functions

If $X$ is a space of finite measure, what is the dual space of $L^p _{loc}$ (the space of locally $p$-integrable functions)? When $p=1$, a good answer has already been provided. What is known for $p ...
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2answers
120 views
+100

Weierstrass's M-test example for uniform convergence and switching Sum and Integral.

How would I go about finding $M_n$ in \begin{equation} \sum_{n=1}^{\infty} \int_{0}^\infty x^{\frac{s}{2}-1}e^{-\pi n^{2}x}dx \end{equation} to show that it is uniformly convergent? UPDATE: Just ...
8
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3answers
261 views

For which $s\in\mathbb R$, is $H^s(\mathbb T)$ a Banach algebra?

According to Theorem 4.39 in Adams & Fournier Sobolev Spaces: If $mp>n$, $m\in\mathbb N$, then $W^{m,p}(\Omega)$ is a Banach algebra, provided that $\Omega\subset\mathbb R^n$ satisfies the ...