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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13 views

4th order correlations of a delta-correlated random process

Say I have a complex random variable A(z) that is $\delta$-correlated, i.e. I have: $ \begin{align}\langle A(z) \rangle &= 0 \\ \langle A(z) A^*(z') \rangle &= \delta(z-z') \end{align} $ ...
3
votes
2answers
128 views

Spectral theorem for representations proof.

Let $H$ be a separable Hilbert space, and $U$ a unitary representation of $\mathbb{Z}^d$ on $H$. Let $\chi_m$ be the characters of the Torus $T^d$, and $m$ the Haar measure on $T^d$. I would like to ...
1
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0answers
18 views

Weak harnack type inequality

I have reached a lemma which I do not have any reference and hint for it. Assume $ \Omega \subset \mathbb{R^N} $ is smooth bounded domain and let $u$ be a positive distributional supersolution ...
1
vote
1answer
24 views

How can we derive the projection formula in general?

The derivation of the well-known projection formula $proj_\vec{b}(\vec{a})=\frac{\vec{a}\cdot \vec{b}}{\vec{b}\cdot \vec{b}}\vec{b}$ uses an argument based completely on geometry. We assume vectors ...
-1
votes
1answer
35 views

The Frechet derivative can be defined in 2 ways. Is there an easy way of showing that they are equivalent?

$F: X \to Y$ ($X$, $Y$ normed vector spaces) then exists a linear transformation $A:X \to Y$ if for every $\epsilon > 0$ exists $\delta >0$ such that $||F(x+h)-F(x)-Ah||\leq \epsilon ||h||$ for ...
2
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0answers
27 views

$C^0([0,1])$ is separable – or isn‘t it?

Using Bernstein polynoms it can be proven that $(X, \|\cdot\|) := (C^0([0,1]), \|\cdot\|_{C^0([0,1])})$ is a seperable vector space. However, here is my “proof” that this space is not seperable: ...
2
votes
2answers
70 views

Best approximation and an inequality

Let $H$ be a Hilbert space. Let $E\subset H$ and $x\notin E$. Suppose that there exists $y^*\in E$ such that $$\|x-y^*\|=\min_{y\in E}\|x-y\|$$ (i.e., $y^*$ is the best approximant of $x$). I hope ...
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0answers
17 views

Uniformly continuous unitary representations.

Let $U$ be a unitary rep. of $\mathbb{R}^d$ on a separable Hilbert space $H$, and $H\cong\oplus L^2_{\mu_v}(\mathbb{R}^d)$ be the spectral decomposition (according to the spectral theorem for these ...
0
votes
0answers
37 views

Dirac Delta Distribution and non-compactly supported test function

I would like to know if there is any problem with defining the following expression: $$ I = \int_0^\infty g(t) \delta(f(t))\mathrm{d}t $$ where $0<\lim\limits_{t\to\infty} g(t) =L<\infty$ and ...
1
vote
1answer
18 views

Limit inferior, weak convergence

I have a question about weak convergence and limit inferior. Let $(X,\Sigma,\mu)$ be a measure space (if necessary $\sigma$-finite measure space). Let $(u_{t})_{t >0}$ be a family of square ...
1
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1answer
38 views

Finding a maximal complete subspace of Riemann Integrable functions on $[0,1]$

I know that the space of Riemann Integrable functions on $[0,1]$ is not complete under the norm $|f|= \int f$. So I was wondering as to what would be a maximal complete subspace of Riemann Integrable ...
0
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2answers
22 views

Existence of unique solution in Banach space

Let $X$ be a Banach space and let $L : X → X$ be a bounded linear operator. Are there situations where $||L||>1$ for which there is a unique solution to $x=Lx+b$? Explain your answer. My attempt: ...
0
votes
1answer
18 views

Dual norm of $L_p$ space

Given $R^n$ is equipped with the norm $||x|| = (\sum_{k=1}^{n} |x_k|^p)^{\frac{1}{p}}$ for some $p ≥ 1$, what is the induced norm on the conjugate (dual) space? I couldn't figure out how to prove ...
0
votes
1answer
30 views

Orthogonal projection in complex Hilbert space

Let $X$ be a complex Hilbert space, and let $T\in L(X, X)$ denote the orthogonal projection onto a closed subspace $M ⊆ X$. (a) Determine the kernel $N(T − λI)$ and the range $R(T − λI)$ of $T − λI$ ...
3
votes
1answer
37 views

How is this the Open Mapping Theorem?

My book has this theorem which it has stated as the Open Mapping Theorem: Suppose X and Y are Banach spaces and $T \in B(X,Y)$ is surjective. Let: $L=\{T(x): x \in X \text{ and } \|x\|\le ...
5
votes
4answers
102 views

An instance of quotient Space $X/M$

Hi Guys i can't find a simple example (with analytic description i mean) that helps me to understand the meaning of quotient space. I've understood the definition ($X$ normed linear space, $M$ closed ...
0
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1answer
31 views

Does the existence of a maximum over $|f_u(x,u(x))|$ imply that $f$ satisfies the Lipschitz condition?

A function $f$ satisfies the Lipschitz condition if: $$|f(x,u_1(x)) - f(x,u_2(x))| \leq A |u_1(x) - u_2(x)|$$ $$\frac {|f(x,u_1(x)) - f(x,u_2(x))|}{|u_1(x) - u_2(x)|} \leq A $$ Does the existence ...
6
votes
0answers
156 views

Regularity of a weak solution

The problem is formulated as follows: Let $\Omega\subset\mathbb{R}^2$ a bounded Lipschitz domain. For $u\in L_2(\Omega)$ one can define the one dimensional weak dirivative $\partial_1 u$. Now define ...
4
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0answers
70 views
+50

What plays the role of the identity for the generalized convolution associated to the Fourier-Bessel transform?

In traditional Fourier theory, the Dirac delta plays the role of an "identity" for the $L^1$ algebra with respect to the usual convolution. The convolution is traditionally built out of group ...
6
votes
1answer
61 views

Do we need completeness for a weak*-convergent sequence to be bounded?

Let $(\phi_n)_n$ be a weak* convergent sequence in the dual of some normed space $X$ with (weak*-)limit $\phi$. If $X$ is Banach then it follows from the uniform boundedness principle that $\sup_n ...
4
votes
1answer
43 views

definitions of order-dense Riesz spaces

I would like to get an intuitive idea as to why the definitions of order-dense and locally order-dense Riesz subspaces were chosen in the following way: A Riesz subspace $F$ of $E$ is order-dense if ...
3
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0answers
33 views

The best constant in Poincare-liked inequality in BV space

Let $\Omega\subset \mathbb R^N$ open bounded smooth boundary. Then we could prove that for any $u\in BV(\Omega)$ and $\omega\in \operatorname{ker}\mathcal E $, where $\mathcal E$ denotes the ...
0
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0answers
50 views

If $\|f\|_X\le c_1\|f\|_Y$ then do we have $\langle f,g\rangle_X\le c_2\langle f,g\rangle_Y$?

Let $X$ and $Y$ be two inner product spaces with inner products $\langle \cdot,\cdot\rangle_X$ and $\langle \cdot,\cdot\rangle_Y$, respectively. Suppose we have $\|f\|_X\le c_1\|f\|_Y$ for any $f\in ...
0
votes
1answer
31 views

Equicontinuous sequence in $C(\mathbb{R^2})$ and Arzela-Ascoli Theorem

Could anyone help with the following problem? I am trying to work out this last practice problem for my Real Analysis prelim but I'm not sure about how to approach it. It looks very similar to the ...
0
votes
2answers
23 views

If $M$ is $F$-measurable, then is it also $F'$-measurable with $F'\subset F$?

$F$ and $F'$ are $\sigma$-algebras, and $M$ is a function from $(\Omega,F)$ to $(\mathbb{R},B(\mathbb{R}))$ If this statement is true, how to reason or understand it in a simple way?
1
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0answers
20 views

The relation of the Homogeneous Sobolev norm and general Sobolev norm

I'm wondering if the inequality $$ \left\| F\right\|_{\dot H^k(\mathbb R^n)} \le C\left\| f\right\|_{L^\infty(\mathbb R^n)} \left\| f\right\|_{\dot H^k(\mathbb R^n)} $$ holds for $k\in[0,10]$ then $$ ...
3
votes
1answer
60 views

Gateaux derivative of $L_p$ norm

For $2\leq p < \infty$, if we consider $f,g \in L_p(X, \mathcal{M},\mu)$ there is the well-known equality $$\frac{d}{dt}\Vert f+tg \Vert_p^p = \frac{p}{2} \int_X \vert f(x)+tg(x) \vert^{p-2} ...
1
vote
1answer
30 views

The intersection of $BV$ space.

Given $\Omega\subset \mathbb R^N$ open bounded smooth boundary. We define $$ TV(u,\alpha):=\sup\left\{\int_\Omega u\operatorname{div}v\,dx,\, v\in C_c^\infty(\Omega;\,\mathbb ...
0
votes
0answers
24 views

What type of self-adjoint operator does $\hat{P}$ has to be for Green's function to result in a radial exponetial $e^{-\| x-t \|^2}$

I was reading the following paper on hyper basis function (HBF) (similar to radial basis function RBF network) and was trying to understand when is it the case that the network has radial basis ...
-1
votes
0answers
18 views

Waveles, a counter example of r-regularity. (Decreasing fast)

I was studying multi-resolution analysis when I found this counter-example that I can't check. Take the space $V_0 := \{ f \in L^2(\mathbb{R}) :supp ~\hat{f} \subset [-\pi,\pi]\}$, where the hat means ...
6
votes
2answers
110 views

Generalized convex combination over a Banach space

The Question: Is the following true? If not, what further hypotheses do I need? Let $X$ be a Banach space, and let $C \subset X$ be closed and convex. Let $P$ be a probability measure over $D$, ...
0
votes
1answer
59 views

Projections: Beppo Levi

Given a Hilbert space $\mathcal{H}$. Consider projections: $$P_\lambda\in\mathcal{B}(\mathcal{H}):\quad P_\lambda^2=P_\lambda=P_\lambda^*$$ And directed indices: ...
2
votes
1answer
30 views

Are invertible linear operators of bounded linear operators also bounded?

I have this definition in my book: Definition: Let X,Y be normed linear spaces. An operator $T \in B(X,Y)$ is said to be invertible if there exists $S \in B(Y,X)$ such that $ST=I_X, TS=I_Y$, ...
3
votes
2answers
92 views

Is $L^1(X) \cap L^2(X)$ a closed subspace of $L^2(X)$ and $L^1(X)$?

Suppose that $X$ be a locally compact Hausdorff space. Could we say that $L^1(X)\cap L^2(X)$ is closed subspace of $L^1(X)$ and $L^2(X)$?
0
votes
0answers
20 views

Is $L^{\infty}(Z)$ first-countable?

Is $L^{\infty}(Z)$ first-countable? A space X is said to be first-countable if each point has a countable neighbourhood basis (local base). $L^{\infty}(Z)$ is dual of $L^{1}(Z)$ with convolution ...
-1
votes
4answers
85 views

Direct proof for convexity of $e^x$ [closed]

Is there any direct proof without using second derivative for convexity of $e^x$?
-4
votes
0answers
42 views

Two topologies are coincide [duplicate]

Let $(X,\tau)$ and $(X,\tau')$ are both metrizable topological vector space, and let $(x_{n})\subset X$ and $x\in X$ and $x_{n}\rightarrow x$ in $\tau$ topology if and only if ...
4
votes
2answers
66 views

Norm of the integral operator in $L^2(\mathbb{R})$.

Suppose we have an integral operator $A$ such that $$Af(x) = \frac{1}{\sqrt{2\pi}}\int\limits_{\mathbb{R}}e^{-\frac{(x-y)^2}{2}}f(y)dy$$ To find $\|A\|$ we can use the unitary Fourier transform $F$, ...
-2
votes
3answers
57 views

Do the topologies with the same convergent sequences coincide? [closed]

Do the same sequential convergence in two topologies the result are the same topologies? Let $X$ be topological space with topologies $\tau$, $\tau'$. Let $(x_{n})\in X$ and $x\in X$ and ...
0
votes
2answers
41 views

what can you say about the solutions of the equation $y' = x^2+y^2$ just by looking at the differential equation

Can we say that the graph is symmetric about origin. Because replacing $x$, $y$ with $-x$, $-y$ does not change the equation Also the slope becomes larger as we move away from origin. Anything else ...
2
votes
0answers
29 views

Equivalent definition of uniform convexity

A Banach space $X$ is said to be uniformly convex if the following is satisfied: For $\epsilon>0, \exists \delta>0$ such that $x,y\in X, \|x\|, \|y\|\leq 1$, $\|x-y\|\geq \epsilon \Rightarrow ...
0
votes
0answers
22 views

A system is complete iff the only functional that zeros its elements is $\varphi=0$

Let $(X,\Vert\cdot\Vert)$ be a normed space. Let $\{x_n\}\subset X$. The dual space $X^\ast$ is the space of the functionals $\varphi:X\to\mathbb{R}$. Prove that $$\{x_n\}\text{ is complete in X}\iff ...
0
votes
1answer
39 views

How can I use Banach Contraction Principle to solve $Ax = b$?

Can anyone explain to me how Banach Contraction Principle (fixed point theorem) makes it easier to solve $Ax = b$?
1
vote
2answers
34 views

How can I compute the infimum of the following non linear functional

I was trying to solve this problem from previous exam of functional analysis and I am stuck Clealy $ \inf_{f \in \mathcal{M}}\phi(f) \leq 0$.(I can choose f=0) If I compute the infimum over the ...
3
votes
1answer
69 views

What are some easy to understand applications of Banach Contraction Principle?

I know that Banach contraction principle guarantees a unique solution to problems of the form $$f(x) = x$$ But for the life of me I cannot understand why this problem is important at all. I don't ...
7
votes
0answers
53 views

Constructing an orthogonal basis with a choice of inner product [duplicate]

Given a linearly independent set of vectors in some vector space, is it always possible to construct an inner product so that the vectors are orthogonal? I know I can construct an appropriate inner ...
4
votes
2answers
94 views

Are the Hermite-Gauss functions linearly dense in $L^1(\mathbb{R})$?

The Hermite-Gauss functions ($t\mapsto H_m(t)e^{-t^2/2}$) are known to be an orthonormal basis for $L^2(\mathbb{R})$, a fortiori linearly dense in $L^2(\mathbb{R})$, and all are in the Schwartz space ...
1
vote
1answer
29 views

Sobolev norm in the definition of Sobolev spaces

I've seen the Sobolev space defined as: The Sobolev space $H^k(\Omega)$ is the set of all functions $u \in L_2(\Omega)$ for which the weak derivative $\partial^\alpha u \in L_2(\Omega)$ for all ...
3
votes
1answer
33 views

Definition of Sobolev space $H^s$ and domain of $-\Delta^s$

The spaces below are on $\partial\Omega$, the boundary of a bounded smooth domain $\Omega$. I read this in the book on page 141. Define $H^2 := \{ u \in L^2 \mid (-\Delta u) \in L^2\}$. And ...
0
votes
1answer
17 views

integral operator with degenerate kernel

Suppose I have an integral operator on $L^2$, $\int_0^1K(s,t)f(t)dt$ where K(s,t) is degenerate. Can I state that the norm of this operator equals its largest eigenvalue absolute value? As a concrete ...