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

How Fourier transform(opeartor) behaves with the operation composition?

It is well-known that Fourier transform takes convolution into a point-wise multiplication, that is, $\widehat{f\ast g}= \hat{f}\cdot \hat{g}$, for $f, g \in L^{1}(\mathbb R).$ That is, bit roughly, ...
1
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2answers
35 views

Characterisation of one-dimensional Sobolev space

I've got some doubts proving that $$H^1_0((a,b))=\{u\in AC([a,b]): u'\in L^2 \text{ and } u(a)=u(b)=0\}:=X.$$Let $$\mathcal A=\{v\in C^2([a,b]):v(a)=v(b)=0\}.$$ $H^1_0((a,b))\subseteq X.$ Let ...
1
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1answer
15 views

Books with exercises or problems on the space of functions of bounded variation

I am studying BV space (the space of functions of bounded variation) by using Evans & Gariepy's book. However, there are no exercises for BV space, and I have no idea where I can find some. ...
2
votes
0answers
25 views

Convex interior topology

I have found a fascinating example of topology on a vector space $V$, but I cannot prove its interesting properties to myself. Let $\mathcal{B}$ be the family of all convex symmetric (i.e. $\forall ...
0
votes
0answers
27 views

Uniform convergence and Weakly convergence

Let $u_{k}\in l^2{\mathbb{(Z)}}$ be a sequence such that for every sequence $n_{k} \in \mathbb{Z}$ the sequence $n_{k}u_{k}\rightharpoonup 0$. Prove that $ u_{k} \rightarrow 0$ in $l^{q}(\mathbb{Z}) , ...
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1answer
33 views

The symetric operator is linear operator. [on hold]

Definition: Let be $X$ unitary space. Operator $A:X\rightarrow X$ called symetric if $$ (Ax\vert y)=(x\vert Ay), (x,y\in X).$$ I saw some books functional analysis but can not find verification of ...
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0answers
20 views

Get locally uniformly convex norm by bounded linear operator

I want to prove this theorem Let $Y$ be a locally uniformly convex Banach space and $X$ be a Banach space and let $T$ be a bunded linear operator from $X$ into $Y$ such that for every bounded ...
3
votes
3answers
618 views

Folland & Functional Analysis

I'm reading Folland's Real Analysis to learn some basic functional analysis. I read through his section Normed Vector Spaces and could make my way through most of the exercises I attempted. I am ...
1
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1answer
104 views

Minimum calculus of variation

Hi I am looking for a criterion that is sufficient to prove that a solution to a functional depending on two functions y(t) and x(t) is an extremum. it is about the following functional$$ \int_0^b ...
0
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0answers
89 views

Why is it hold for types of operators?

We ‎let ‎the ‎state ‎space ‎be‎ ‏‎‎‎$ \mathcal{H} =‎ ‎‎H_{E}^{2}(0 , 1) \times L^2(0 , 1) $‎ equipped with the norm ‎ \begin{align} \| (f , g) \| = \int_{0}^{1} [ |f''(x)|^2 + |g(x)|^2] ‎\mathrm{d}x‎ ...
5
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1answer
47 views

Existence of integral equation solution

I am trying to prove a differentiable solution in some open interval about the origin for the equation: $$u(x) + u(x)^2 + \int_0^x (1+\cos(x+u(y))) dy = 0$$ I have been trying to prove it as a ...
2
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0answers
35 views

Modulus of convexity

The modulus of convexity of a normed linear space $X$ is defined as $$ \delta(\epsilon) = \inf \left\{ 1- \frac12 \|x+y\|: x,y \in \overline{B_X}, \|{x-y}\| \geq \epsilon \right\} $$ or as $$ ...
0
votes
0answers
30 views

Formal definition of “node” with respect to eigenvalues and functional analysis.

I'm concerned with a special problem of spectral analysis for a certain Sturm-Liouville-differential-operator, that is to say $L:=\frac{d^2}{dx^2}-q(x)$ and the spectrum $\sigma(L)$. While reading an ...
0
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1answer
38 views

Difference between Half Quadratic vs Quadratic

Half quadratic minimization/penalty/optimization, I am unable to find any related material/resources. If anyone can point to some useful resources, it will be great
4
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2answers
129 views

Extensions and Kazhdan's Property (T)

Is Kazhdan's property (T) stable under extensions? i.e. if $G$ is an extension of a group with property (T) by a group with property (T), does it follow that $G$ has property (T)?
6
votes
3answers
212 views

Dual norm intuition

The dual of a norm $\|\cdot \|$ is defined as: $$\|z\|_* = \sup \{ z^Tx \text{ } | \text{ } \|x\| \le 1\}$$ Could anybody give me an intuition of this concept? I know the definition, I am using it ...
6
votes
1answer
252 views

Is this proof good? Identifying extreme points of the unit ball in a function space

I want to prove: If $K$ is compact $T_2$ then the extreme points of the unit ball of $C(K)$ are precisely the functions $f\in C(K)$ such that $|f(k)|=1$ for all $k\in K$. Here is my proof: Can someone ...
4
votes
2answers
62 views

Example of a function $u\in L^\infty(0,T,H^1)$ such that $u_t\notin L^\infty(0,T,H^1)$

Could someone give me an example of a function $u\in L^\infty(0,T,H^1)$ such that $u_t$ exists (in the distributional sense), $u_t\in L^\infty(0,T,L^2)$ and $u_t\notin L^\infty(0,T,H^1)$? Thanks. ...
1
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1answer
42 views

Characterizing C* algebra generated by elements.

Let $A$ be a C*-algebra and $A_0\subset A$. Then it is known that the $\mathbb{C}$-algebra generated by $A_0$ (i.e. the intersection of all sub-$\mathbb{C}$-algebras containing it ) is just the ...
6
votes
2answers
315 views

Unitisation of $C^{*}$-algebras via double centralizers

In most of the books I read about $C^{*}$-algebras, the author usually embeds the algebra, say, $A$, as an ideal of $B(A)$, the algebra of bounded linear operators on $A$, by identifying $a$ and ...
2
votes
2answers
37 views

If $f(x)g(y)$ is a measurable function, and $f$, $g \in L^{1}(dm)$, does this imply $g(y - x) \in L^{1}(dm)$?

Question rephrased Suppose we are working in $(\mathbb{R}, \Sigma(m^{*}) \times \Sigma(m^{*}), m \times m)$ where $m$ is Lebesgue measure. Note that our $\sigma$-algebra is not necessarily complete. ...
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1answer
40 views

Orthogonal basis of complete Euclidean space

friends! I read that any complete Euclidean, complex or real, space $R$ has a (normalized) orthogonal basis. By orthogonal basis an orthogonal system of vectors such that the smallest closed subspace ...
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0answers
39 views

Calcul of limit [on hold]

What is the limit of $$\lim_{f \rightarrow 0} \frac{ \nabla {f(x)} }{\sin{(f(x))}}?$$ We can use the Poincare inequality and the famous limits: $$\lim_{x\rightarrow 0} ...
1
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1answer
52 views

Explaining why $\Vert x_n\Vert _1=\frac{3}{4}$

Let $X_1$ respectively $X_2$ denote the space $C [a, b]$, $a <b$ with norm: $$\Vert x\Vert_1=\int_a^b \vert x(t)\vert dt; $$$$\Vert x\Vert_2=\left(\int_a^b \vert x(t)^2\vert ...
2
votes
0answers
57 views

$ d(x,y)^2 \le d(x,z)^2 - d(y,z)^2\color{}{+\varphi\big(x,y,z,d(x,y),d(y,z) \big)}? $

In a metric space $(M,d)$ the triangle inequality $d (x, z) \le d(x, y) + d (y, z)$ gives us's the inequalitie $$ \quad d(x,y)^2 \ge d(x,z)^2 - d(y,z)^2\;\color{}{{-2\cdot d(x,y)\cdot d(y,z)}} $$ ...
2
votes
1answer
35 views

If $T: X \to Y$ is norm-norm continuous then it is weak-weak continuous

Let $X,Y$ be normed linear spaces (or Banach spaces if necessary) and let $T: X \to Y$ be linear. We call $T$ norm-norm continuous if $X,Y$ are endowed with the norm topology and similarly, weak-weak ...
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votes
0answers
14 views

Convex Inequality describing Functions inside specific area

Let us assume that we have two functions $f_1$, $f_2:[0,1] \rightarrow \mathbb{R}^{2}$, which describe each a point trajectory on the plane. Let us further assume that we parametrize those functions ...
2
votes
0answers
23 views

Resolvent and spectrum of a self-adjoint extension

In this paper, they give the resolvent, spectrum, and eigenfunctions of the self-adjoint extension of the Laplacian on a rectangle that corresponds to a delta potential at an arbitrary point (items ...
2
votes
1answer
274 views

Characterization of compactness in weak* topology

Let $ X $ be Banach space, and $X^*$ its dual. A set $ F \subset X ^ * $ is weakly-* compact if and only if $ F $ is closed in the weak* topology and is bounded in norm. How does one prove this ...
0
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1answer
48 views

Question about statement of Fubini's theorem

This is a question on the statement of Fubini's theorem for measurable sets. The theorem looks like this: Let $(X \times Y, \overline{\Sigma \times \tau}, \lambda = \mu \times \nu)$ be a complete ...
2
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0answers
40 views

How to solve this finite-difference equation?

How to solve the following finite-differences equation: $$f(x) = f(x-1) + f(x-\sqrt{2}), \quad x\in [\sqrt{2}, +\infty) \,?$$ Let's say $f(x) = f_0(x)$ for $x \in [0, \sqrt{2})$ is a given function. ...
0
votes
1answer
46 views

General solution of a finite-difference equation with real non-commensurable differences.

How to find a general solution of the following functional (recurrence) equation: $$f(x) = c_1 f(x - a_1) + \dots + c_n f(x - a_n), \tag{1}$$ where $c_i, a_i$, $i = 1, ... n$ are arbitrary real ...
2
votes
1answer
24 views

Find the values of $p$ for which the given sequence converges in $l^p$ norm or weakly

We consider $E_p=(c_{00}, ||\cdot||_{p})$ (where $c_{00}$ are the sequences that are zero except in a finite number of values and $1\leq p \leq \infty$) and the sequence:$$(x_n)_{n\in ...
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votes
0answers
34 views

Book suggestion functional analysis [duplicate]

I am studding functional analysis and applications. Does anyone have a good recommendation of books//lectures/resources/etc.? Thanks.
0
votes
0answers
25 views

Proving concavity for complicated function

I have a rather complicated function, $f$, that I am trying to demonstrate is log-log-concave, i.e., $$\frac{d^2\log f}{(d\log x)^2}\leq 0.$$ The reason I think it is concave is purely heuristic. ...
0
votes
1answer
31 views

Dense subset of $L^{2}$ such that $x^{-1/2}f \in L^{1}$ and $\int_{[0, 1]}x^{-1/2}f\, dx = 0$

Does there exist a dense set of functions $f \in L^{2}([0, 1])$ such that $x^{-1/2}f(x) \in L^{1}([0, 1])$ and $\int_{0}^{1}x^{-1/2}f(x)\, dx = 0$? I've noticed that $\int_{0}^{1}x^{-1/2}f(x)\, dx = ...
1
vote
2answers
56 views

$2\pi$ in the Definition of Fourier Transform

Most textbooks I read define Fourier transform of a function $f \in L^2(\mathbb R)$ as $$ \hat f (\xi) := \int_\mathbb R f(x) e^{-2\pi i x \xi} dx. $$ However, in class my teacher defines it without ...
2
votes
1answer
26 views

Kernel of the Extension of a Bounded Linear Operator

Suppose $T\colon E\to F$ is a bounded linear operator between Banach spaces. Moreover let $i\colon E\to E’$ be a dense, compact inclusion of $E$ into some other Banach space $E’$. Finally assume that ...
10
votes
2answers
1k views

The direct sum of two closed subspace is closed? (Hilbert space)

I know that if $X$ is a Banach space, then, the direct sum of two closed subspace $X_1$ and $X_2$ is not necessarily closed. But what if $X$ is Hilbert? I assume there is something to do with the ...
1
vote
2answers
58 views

Eigenvalues of symmetric elliptic operators

As stated in one of my previous questions already, I have not had so much exposure to theoretical linear algebra. This time, I'm reading a theorem and proof from PDE Evans, 2nd edition, pages 335-356. ...
13
votes
1answer
220 views

Injectivity of the operator $(Ax)(t)=\int_0 ^1 k(s,t) x(s)ds$

Let $X=C([0,1],\mathbb{R})$ (equipped with the supremum norm). Let $A$ be the operator defined for each $x\in X$ by $$(Ax)(t)=\int_0 ^1 k(s,t) x(s)ds,$$ where $k:[0,1]\times [0,1]\to \mathbb{R} $ is ...
2
votes
1answer
51 views

Showing a certain operator on a set of holomorphic functions attains its supremum

Let $G \subseteq \mathbb{C}$ be open and connected, and fix $a_1, a_2 \in G$. Let $\mathcal{F} := \{f: G \rightarrow \mathbb{C}: f$ is holomorphic and $|f(z)| \leq 1 \}$. Define $\Phi: ...
0
votes
3answers
43 views

If $a_n\ge nb_n$ and the sequence $(b_n)$ is unbounded, then the differences $a_{n+1}-a_n$ are also unbounded

Let $(a_n)_{n\geq 1}$ and $(b_n)_{n\geq1}$ be sequences of positive numbers such that $a_n\geq n b_n$ for all $n >1$. Prove that if $(a_n)_{n\geq 1}$ is increasing and $(b_n)_{n\geq 1}$ is ...
2
votes
1answer
45 views

Intersection of nested closed bounded convex sets in Euclidean space

I read that in a complete Euclidean space - i.e. a normed real space with the norm induced by the scalar product - any sequence of nested bounded non-empty closed convex sets has a non-empty ...
0
votes
1answer
45 views

Boundedness of continuous functions on compact sets

Let $E$ and $F$ be two metric spaces. If $K$ is a compact subset of $E$ then a continuous function $f:K\to F$ is always bounded and reachs its maximum. What happens if we replace $K$ by a closed ...
3
votes
1answer
35 views

If $(X \times Y, \overline{\Sigma \times \tau}, \mu \times \nu)$ is $\sigma$-finite, does that imply $(X, \Sigma, \mu)$ is $\sigma$-finite?

I'm having trouble proving or disproving the statement: If the product space $(X \times Y, \overline{\Sigma \times \tau}, \mu \times \nu)$ is $\sigma$-finite, then so is $(X, \Sigma, \mu)$. I ...
0
votes
1answer
47 views

Borel Measure on Banach Space

While thinking about what some measure on an infinite dimensional Banach space could look like a came across the point that if I'd like to assign a size to all epsilon balls, they by Riesz' lemma ...
1
vote
1answer
49 views

Condition on the kernel of the integral operator to belong to the trace class?

Let $\mu$ be a finite compactly supported Borel measure on the real line. Consider the integral operator $K$ on $L^2(\mu)$, $$ (Kh)(x)=\int h(y)k(x-y)\, d\mu(y), $$ where $k$ is a fixed function. ...
-1
votes
1answer
43 views

Book suggestion geometry of Banach spaces

I am studding geometry of Banach spaces and applications in metric fixed point theory. Does anyone have a good recommendation of books//lectures/resources/etc.? Thanks.
2
votes
1answer
36 views

If a compact operator satisfies $T^nx\to0$ weakly for all $x$, then $\|T^n\|\to0$

Let $H$ be a real Hilbert space, $T:H\to H$ be a compact operator. Suppose that for every $x\in H$, sequence $(T^n x)_{n\in \mathbb{N}}$ converges weakly to $0$. How to prove that $ ...