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, ...

learn more… | top users | synonyms (1)

0
votes
0answers
5 views

Functional optimization: maximize a double integral where the functional appears twice

Please help me solve the following optimization problem. Suppose that you have to choose a function $U: [0,1]\mapsto [0,1],$ which must be nondecreasing ($U'\geq 0$) to maximize the following ...
1
vote
1answer
14 views

Boundedness of smooth functions approximating an Lp function

We all know that the space of smooth functions on Euclidean space with compact support is dense in the Lp spaces, for p strictly less than infinity. Now my question is: suppose there is a function f ...
0
votes
0answers
21 views

Linear Independence for functions defined by integration

I came across this problem while doing some work. I'm been unable to make any progression on it. Any suggestions would be greatly appreciated. Given that the set of strictly positive functions ...
2
votes
0answers
15 views

Example of compactness James' Theorem

I would like to find any example on any known space of application of the classical compactness James' Theorem for obtaining a good visualitation of how this result works. For instance, on any space ...
-1
votes
0answers
14 views

Infinite vector spaces which aren't reflexive? [on hold]

Find an example of a) countable b) uncountable vector space which is not reflexive?
4
votes
1answer
16 views

Going from composites to individual functions

$f(g(k(x)))=\sqrt{1+4x^2}$ and $g(k(f(x)))=1+4x$ What is a systematic way to solve for $f$,$g$,and $k$? I never learned anything like this in algebra.
3
votes
0answers
21 views

Densely defined bounded integral transforms on $L^2(\Bbb R)$

This is a question I've contemplated for quite some time since it's pretty closely related to Fourier theory (particularly choosing the "right" space to define the Fourier transform on). However I've ...
1
vote
1answer
20 views

Proving an identity of the resolvent set

$\{T(t)\}_{t\ge 0}$ is a $C_{0}$-semigroup with infinitesimal operator $A$. I'm trying to prove that the set $\{ z|\text{ Re }z>\omega_{0}\}$ belongs to $\rho(A)$, and for $z$ in this set, the ...
8
votes
1answer
81 views

What exactly is an integral kernel?

I am not sure if I have seen integral transforms in the right way, but given a transform like fourier transform - its actually a basis transformation right ? $$ F(y) = \int K(x,y) f(x) \text{d}x $$ ...
1
vote
0answers
48 views

sign of eigenvalue [on hold]

Let L is a linear operator, we say it is stable if it has negative spectrum, why it is equivalent with there exists some $‎\varepsilon‎>0$ such that $$\langle Lh,h \rangle ...
0
votes
1answer
18 views

Weak convergence of bounded nets

Let $(x_\alpha)_{\alpha\in A}$ be a bounded net in $c_0$. For all $\alpha\in A$, let $x_\alpha = (x_\alpha^n)\in c_0$; if, for every $n\in\mathbb{N}$, $(x_\alpha^n)_{\alpha\in A}$ is a net that ...
0
votes
0answers
23 views

Norms on $L(V,W)$

Let $V,W$ be normable topological vector spaces over $\mathbb{F}$. Let $C(V,W)$ be the set of continuous linear transformations $T:V\rightarrow W$. Let $||\cdot||_V, ||\cdot||_W$ be norms on $V,W$ ...
1
vote
1answer
26 views

Choosing a smooth function with desirable properties

Consider a smooth function $\varphi \in C^\infty[0, 1]$, where $\varphi (1) = 0$. My question is, can we necessarily choose another function $\psi \in C^\infty[0, 1]$, such that $\psi \geq 0, \psi(1) ...
0
votes
1answer
24 views

Fourier transform is unitary proof and other unitary integral operators

There is this old unanswered question: Proof the Fourier Transform is Unitary/Not Unitary What is the easiest way to see that the Fourier transform is unitary and why it is important to have constant ...
1
vote
0answers
23 views

A small doubt regarding a previously asked limit of convolution

Previously, I asked this question to the forum. Pointwise limit of convolution Now, a question in this regard is coming to my mind. Suppose, we don't have the integral; i.e. we have the ...
8
votes
1answer
42 views

A sequence that converges weakly but not in the Cesàro sense

Let $H$ be a Hilbert space over $\mathbb{C}$ with inner product $\langle\cdot,\cdot\rangle$, and let $\{x_n\}_{n=1}^\infty\subseteq H$, $x\in H$. I'm using the following definitions: ...
0
votes
0answers
31 views

Is embedding a function make sense?

The embedding is defined in the wikipedia link as "In mathematics, an embedding (or imbedding [1]) is one instance of some mathematical structure contained within another instance, such as a group ...
2
votes
1answer
44 views

Difficulty in understanding converse part of proof of a propostion in Andrew Browder's Mathematical Analysis

Proposition: Let $\mu$ be finitely additive set function, defined on the algebra $\mathscr A$. Then $\mu$ is countably additive if and only if its has following property: if $A_n \in \mathscr A$ ...
0
votes
0answers
22 views

Non symmetric operator

My question is: there is some kind of mathematical theory that allows one to prove selfadjointness for an operator which is not initially defined as a symmetric operator? To prove that some operator ...
2
votes
0answers
17 views

Measure-preserving map between a function and its symmetric rearrangement

Let $f \, \colon \mathbb{R}^d \rightarrow[0, \infty)$ be a function such that the sets $ \{ y \: \colon f(y) > \lambda \}$ are of finite Lebesgue-measure, for every $\lambda \geq 0$. Then, we can ...
1
vote
0answers
26 views

Finding the norms of an Operator (under different norms)

Let $X := C([a, b])$ and $k(· , ·) \in C([a, b])\times C([a, b])$. Let $$\alpha_1 := \sup_{ t\in [a,b]} \int_{a}^{b} |k(s, t)|ds$$ and $$\beta_1:= \sup_{s \in [a,b]} \int_{a}^{b} |k(s, t)|dt$$ Define ...
0
votes
1answer
25 views

Approximation of characteristic function by mollifiers

I have been asked to show that the Heaviside function $H := \chi_{[0,+ \infty)}$ does not admit weak derivative in $L^1_{loc}(\mathbb{R})$. Here's my reasoning: By definition the weak derivative of ...
1
vote
1answer
14 views

checking definition of bounded linear function involves operator maps between different spaces

Let $H$ and $K$ be two Hilbert spaces. Let $T:K\to H$ be a bounded linear operator. Denote the inner products on $H$ and $K$ by $\langle\cdot,\cdot\rangle_H$, $\langle\cdot,\cdot\rangle_K$. Fix any ...
0
votes
0answers
12 views

An elementary inequality in the context of Strictly Convex Function

Suppose: whenever $\epsilon \gt 0$ , define: $\zeta (y, \epsilon) = \frac{\eta (y+\epsilon) - \eta (y)}{\epsilon} - \eta'_{+}(y)$ ; where: $\eta$ is STRICTLY CONVEX CONTINUOUS FUNCTION & ...
0
votes
2answers
45 views

Is this infinite-dimensional normed linear space a complete one?

Let $V=\{(x_1, x_2, ...)|x_i\in\mathbb{R}, i=1, 2, ...\}$ be a space of infinite-dimensional vectors. For each $x\in V$, define the norm $\lVert x\rVert =\sum_{i=1}^\infty \lvert x_i\rvert$. If the ...
0
votes
0answers
22 views

Completeness of metric, normed and inner product spaces

For a metric space to be complete, it needs to have all cauchy sequences converge in the metric. 1) For a normed space to be complete, does it need Cauchy sequences to converge in the norm or in the ...
1
vote
1answer
31 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
votes
1answer
32 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 ...
1
vote
1answer
51 views

Why is $\overline{\operatorname{span}\{e_n\mid n\in\mathbb{Z}\}}=L^2(\mathbb{T})$?

I want to know, why $\{e_n\mid n\in\mathbb{Z}\}$ is an orthonormal basis of $L^2(\mathbb{T})$, where $\mathbb{T}=\{z\in\mathbb{C}\mid |z|=1\}$, $e_n(z)=z^n$, and $\int_{\mathbb{T}} ...
1
vote
1answer
17 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 $c_{00}$ ...
2
votes
1answer
38 views

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 ...
1
vote
1answer
22 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 ...
0
votes
1answer
43 views

The highest direction of the trace operator

Let $W$ be a real and symmetric matrix $m \times m$ from the set $M_{m,m}$, and $T:M_{m,m} \rightarrow \mathbb{R}$ a function defined by $T(W) = \operatorname{trace}(W^3)$. We are interested to find ...
0
votes
1answer
44 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 ...
1
vote
1answer
53 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
votes
1answer
50 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 ...
2
votes
3answers
41 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
votes
1answer
52 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
votes
0answers
32 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 ...
0
votes
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 ...
0
votes
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 ...
3
votes
0answers
41 views

some important proofs about adjoint operators [duplicate]

I was told that the formal adjoint of the gradient is the negative divergence. Let $A : H\to H$ be a bounded, linear operator, The adjoint of $A$, i.e. $A^*: H\to H$ satisfies \begin{equation*} ...
0
votes
2answers
26 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 ...
2
votes
1answer
20 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 ...
1
vote
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 ...
0
votes
0answers
41 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 $x$. ...
3
votes
0answers
41 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: $$\cos\left(t^2\right) = \int_{-\infty}^{\infty} e^{itx}f(x)\,dx$$
3
votes
1answer
53 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 ...
1
vote
1answer
30 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 ...
1
vote
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 ...