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
1answer
12 views

Regarding uniform and pointwise convergence

If a real sequence $(f_n)$ of functions converges to a function $f$ uniformly over a domain $D$ except at a a finite amount of points $x_1,\cdots,x_k$, but it happens that at each of these points, ...
-3
votes
0answers
20 views

Show that the adjoint operator of the zero operator is zero

someone can help me on this issue, demonstrate the following , 1) the operator adjoint operator of zero is zero operator. 2) the operator adjoint operator identity is the identity operator.
1
vote
0answers
16 views

How would we generate a basis of sigmoidal functions?

I am trying to figure out how to generate a basis of sigmoidal functions. My issue is thus: there are several possible generating functions for a sigmoid (logistic curves, error functions, arctangent, ...
-3
votes
1answer
46 views

can anyone help me with following question attached in image file [on hold]

Let $(X,\|\cdot\|)$ be a normed space, where $$X=\{(a_n)_{n\geq 1} \mid (a_n)_{n\geq 1} \text{, bounded real sequence}\}$$ and $$\|(a_n)_n\|=\sup_{n\in N} |a_n|$$ Let $$ M=\{(a_n)_n\in X\mid 0\leq ...
1
vote
0answers
21 views

Prove that norm of Yosida approximations blows up

I need an help with the following exercise about Yosida approximations. Let's fix the notation: let $A$ be an unbounded linear operator define on a dense domain $D(A)\subset X$ of a Banach space $X$. ...
0
votes
1answer
11 views

Trace theorems for arbitrary differentiability $k$, with embedding constants under control as $k\to\infty$

The usual trace theorem (with non-optimal exponents, but I don't care for those at the moment) says that $$ W^{1,p}(\Omega)\hookrightarrow L^p(\partial\Omega) $$ for Lipschitz domains. When ...
0
votes
1answer
9 views

What is the mean value theorem for the Fréchet (total) derivative?

What is the mean value theorem for the Fréchet (total) derivative? Off the top of my head, it's something like $$ \|F(x+h)-F(x)\|\leq \sup_{c\in[0,1]} \|F^\prime(x+ch)\|\|h\| $$ but the double ...
0
votes
2answers
35 views

Could someone explain me the task?

this is the question: Show that for each linear map $f:\mathbb R^d → \mathbb R^e$ there exists $a < \infty$ so that $\|fw\|< a\|w\|$ for each $w$ in $\mathbb R^d.$ And my problem is that $f$ ...
0
votes
2answers
17 views

Predual of $l^1(\Gamma)$

Let $\Gamma$ be an uncountable index set. For example $\Gamma=\mathbb R$. Let $l^1(\Gamma)$ be the set of functions with countable support and finite sum: $$ \sum_{a\in\Gamma}|f(a)|<\infty. $$ The ...
2
votes
2answers
173 views

Laplace operator defined on a Sobolev space

Consider the Laplace operator $$A:W^{2,2}(\mathbb{R})\to L^2(\mathbb{R})\;\;\\A u = -u^{\prime \prime}$$ I want to know why this operator is closed (I'm using the closed graph theorem): Let ...
0
votes
1answer
16 views

Is $C^\infty_0(\Omega)$ complete with the norm $\|u\|_\Delta:=(\|u\|_{L^2(\Omega)}^2+\|\Delta u\|_{L^2(\Omega)}^2)^{1/2}?$

Let $\Omega$ be an open subset of $\mathbb R^n$. Is it true that $C^\infty_0(\Omega)$ is complete with the norm $$\|u\|_\Delta:=(\|u\|_{L^2(\Omega)}^2+\|\Delta u\|_{L^2(\Omega)}^2)^{1/2}?$$ Above ...
0
votes
0answers
15 views

convexity of a function of 2 variables

$f\colon\mathbb{R}\to\mathbb{R}$ is continuous and $|f|$ is convex. Prove that $F\colon\mathbb{R}^2\to\mathbb{R}$ defined as $F(x,y)=|f(x)|+|y-f(x)|$ is convex.
1
vote
0answers
23 views

Algebra with element having empty spectrum?

The definition of the spectrum makes sense for any algebra. I guess we can go to the unitization to make sense of it even non-unital algebras. Recalling the well-known fact that for normed algebras, ...
1
vote
1answer
26 views

Density of a subset of a Hilbert space

I've been trying with a colleague but we could not come to a solution. The problem is as follows: Let $M$ be a subset of a Hilbert space $H$, and let $v,w\in H$. Suppose that $\langle ...
1
vote
2answers
28 views

show that if $y$ is orthogonal to $x_n$ and $x_n$ converges to $x$ then $x$ is orthogonal to $y$

help me. someone who can help me? spaces is inner product. It is section 3.2, issue 4 introduction to functional analysis book author Kreyszig
4
votes
1answer
36 views

Inverse Function Theorem for Banach Spaces

In the middle of a proof of the Inverse Function Theorem (namely, the proof of Baby Rudin), we use the fact that if $A$ is invertible and: $$ ||B-A||~||A^{-1}|| <1$$ then $B$ is invertible. The ...
0
votes
1answer
44 views

The set of infinite sequences with finitely many nonzero values is dense.

Could I get a proof to this lemma or a reference if a proof is too time consuming?
2
votes
2answers
32 views

Continuity and differentiability of $f(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{n!(x+n)}$

Given the series: $$\sum_{n \ge 0} \frac{(-1)^n}{n!(x+n)}$$ Let $f_n(x)$ denote its general term. Let $f(x)$ denote its sum (when exists). The question asks to: $i)$ Find the domain $\mathbb D$ on ...
0
votes
0answers
20 views

Strongly continuous semigroup Kolmogorov forward integral equation

Let $\{ P_t \}_{t \geq 0}$ be a SCSF($\mathcal{S}$) (strongly continuous semigroup on $\mathcal{S}$) on the space $(E,\mathcal{E})$, where $E$ is a Polish space, equipped with the ...
-3
votes
0answers
14 views

Hilbert space and Orthogonal complement [on hold]

Let $M$be a linear subspace of Hilbert space $H$. Show that $$cl M = H \Leftrightarrow M^{\perp} = \left\{ 0 \right\}$$
1
vote
2answers
27 views

“isomorphic” normed spaces and reflexivity

Let X, Y be normed spaces and suppose that there exists an bijective isometry between them. And if X is reflexive, then it is intuitively clear that Y is reflexive also. But, when I tried to prove ...
0
votes
0answers
42 views

what would be the formula of $\phi$ in this question?

Suppose $\phi:\mathbb{C}^2\longrightarrow\mathbb{C}^2$ be an entire map (i.e, the components of $\phi$ are entire in each variable separately) with $\phi_1$,$\phi_2$ as its components satisfying ...
0
votes
0answers
25 views

Example of the inequality $c_0\neq\bigcup l_p$

As part of an exercise, I was asked to prove or disprove the following proposition: There exists an $x\in c_o$, such that $x\notin l_p$ for every $1\le p\lt\infty$. Before I show my proof, I will ...
0
votes
0answers
13 views

convex function with Hessian measure $D^2 f \leqslant \lambda$ $\lambda$-concave?

Let $f:R^n \to R$ be convex (may not $C^1$), $$[D^2f]=[D^2f]_{ac}+[D^2f]_s=[h_{ij}] L^n+[D^2f]_s$$ is the Lebesgue decomposition of the Hessian matrix. Where $[h_{ij}]$ is the density w.r.t the ...
2
votes
1answer
35 views

Proof Norm is Continuous

Someone just asked me why the norm of a normed space is continuous, and the answer I gave them satisfied them, but I'm not sure if it should. Something seems amiss. Let $\rho: X \to \mathbb{R}^+_0$ ...
0
votes
0answers
43 views

Is the linear operators must be invertible to from a category?

I am trying to understand the concept of category in mathematics. For example the following link talks about category $Lin$ which is an Abelian category. ...
0
votes
1answer
31 views

$L^1([0, 1]) \subset C([0, 1])^*$

Basically my question is: how can I prove that $L^1([0, 1]) \subset C([0, 1])^*$, where $C([0, 1])$ represents all continuous functions on $[0, 1]$, and the superscript $^*$ means the dual space. ...
2
votes
0answers
20 views

An extension of Kato's Selection Theorem?

One formulation of the well-known Kato Selection Theorem states that, given an analytic family of $n \times n$ complex, symmetric matrices $M(t)$, one can choose an orthonormal basis $\{e_i(t)\}_{i = ...
1
vote
1answer
29 views

If $f \in L^2 \cap C_c$ then $\sum_{n \in \mathbb{Z}}|\hat{f}(\xi+2\pi n)|^2 = a_0+…a_n \cos(2 \pi n \xi)$

Let $f \in L^2 \cap C_c$ , then I want to show that $$g(\xi):=\sum_{n \in \mathbb{Z}}|\hat{f}(\xi+2\pi n)|^2 = a_0 + 2 \sum_{n=1}^{N}c_n \cos(2\pi n \xi) $$ for some $N \in \mathbb{N}.$ Does ...
5
votes
0answers
77 views

Difficult Fourier transform exercise

I am currently dealing with a problem in functional analysis where I want to show the following. Let $\phi_{k+1}(x):=\sqrt{2} \sum_{n \in \mathbb{Z}} h(n) \phi_k(2x-n)$ and $\phi_0= \chi_{[0,1]},$ if ...
-1
votes
1answer
58 views

Example of projection sequence on Hilbert space with strong limit P

Let $P_n$ be strongly convergent with limit $P$, where $P_n$'s are projections on a Hilbert space $H$.Suppose that $P_n(H)$ is infinite dimensional. Show by example that P(H)$ may be finite ...
3
votes
1answer
51 views

Folland, “Real Analysis”, Chapter 5.3, Exercise 36.

Folland, "Real Analysis", Chapter 5.3, Exercise 36: Let $\mathcal{X}$ be a separable Banach space and let $\mu$ be counting measure on $\mathbf{N}$. Suppose that $\left\{x_n\right\}_1^\infty$ ...
0
votes
1answer
32 views

Invertible , bounded linear operator on a Hilbert space

Suppose we have an invertible, bounded linear operator $K$ on a Hilbert space $H$. Is there a constant $c \in \mathbb{R}_+$ such that $$ ||Ku|| \geq c||u|| $$ for all $u \in H$ ?
2
votes
0answers
27 views

Prove that disk algebra is isomorphic to the closure of $\mathbb{C}(z)$ in $C(\mathbb{T})$.

Let $D = \{ z \in \mathbb{C} : |z| < 1 \}$ be the open unit disk and $\mathbb{T} = \{ z \in \mathbb{C} : |z| = 1 \}$ its boundary. We will naturally write $\bar{D}$ for its closure $\{ z \in ...
0
votes
1answer
19 views

Prove that Euler's equation can be written in a specific form

According to my notes, the following theorem holds: If $y$ is a local extremum for the functional $J(y)= \int_a^b L(x,y,y') dx$ with $y \in C^2([a,b]), \ y(a)=y_0, \ y(b)=y_1$ then the extremum $y$ ...
-1
votes
0answers
22 views

Prob. 8, Sec. 3.2 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS [on hold]

if $\|x+ \alpha y\|\ge \|x\|$ for all $\alpha$, then show that $x$ and $y$ are perpendicular. If you could explain this I shall be thankful....
4
votes
0answers
37 views

Spectral theorem for a pair of commuting operators

Let $H$ be Hilbert space and $A$, $B$ - self-adjoint (bounded or unbounded) operators on $H$. According to spectral theorem for every bounded Borel function $f: \mathbb{R}\to \mathbb{R}$ we have ...
0
votes
0answers
17 views

Why $q(]-1,1[)=B(q,0,1)$.

Let $(X,\mathcal T)$ a topologique space defined by $\Gamma$ a set of semi norm. Prove that if $q:X\to \mathbb R$ is a continuous semi norm, then $$\exists p\in\Gamma_f,\exists \alpha>0:q\leq ...
2
votes
1answer
21 views

Fourier transform of $L^1$ function square summable?

It is known that for a $L^1$ function $f: \mathbb{R} \rightarrow \mathbb{C}$ the Fourier transform vanishes at infinity and is continuous. Does this even mean that $(\hat{f}(n))_{n \in \mathbb{Z}}$ is ...
1
vote
1answer
27 views

Problem with the Definition of contractible set

I have this definition of contractible set: we say that $A\subset X$ is contractible in $X$ if there exists a continuous function $\eta:[0,1]\times A\rightarrow X$ such that $\eta(0,x)=x, \forall ...
3
votes
0answers
11 views

The relation between the (algebraic) dimensions of a normed linear space and its dual.

What is the relation between the (algebraic) dimensions of a normed linear space and its dual, for example can we say $\dim X \leq \dim X^*$, for a normed linear space $X$?
1
vote
0answers
13 views

Unitary operator with absolutely continuous spectrum

I have a unitary operator $U_0$ acting on $H := L^2(\mathbb{T}=[-\pi,\pi]; \mathbb{C})$, denote its spectral family by $\{ E_0(\cdot) \}$. Moreover, the spectrum of $U_0$ is purely absolutely ...
0
votes
0answers
36 views

Show $\lim_{n\to\infty}(Lu_n) = L(\lim_{n\to\infty} u_n)$

Suppose $\{u_n\}$ is a convergent sequence in Hilbert space $H$ and $L$ is a bounded (continuous) linear operator on $H$. Use the definition of convergence to show that $\lim_{n\to\infty}(Lu_n) = ...
1
vote
0answers
33 views

Find eigenfunctions of the integral operator with kernel $\sum\limits_{n=1}^\infty \frac{1}{n^2} \sin((n+1)x)\sin(ny)$

Find the eigenvalue and eigenfunctions of the integral operator $Ku=\int_0^\pi k(x,y)u(y)dy$. $k( x,y) = \sum\limits_{n=1}^\infty \frac{1}{n^2} \sin\big((n+1)x\big)\sin(ny)$. This is how I ...
0
votes
1answer
24 views

Orthogonal set of a set in Hilbert space

This is an exercise in the Folland Real Analysis p.177. I first thought it is an easy one, but it turns out to be a lot trickier..... I have no idea how to deal with the so-called "double ...
0
votes
0answers
19 views

Is it possible to modify norm of Sobolev space suitable for ill-posed problems.

I have trying to pose my problem mathematically for quite some time now. I am not sure even if I am close to defining properly. Would anyone please help: I am interested to study ill-posed problem of ...
1
vote
1answer
40 views

Countable set in a Banach space which spans densely?

Let $\mathcal{C}(\mathbf{T})$ be the algebra of continuous complex functions on the unit circle $\mathbf{T}$. Consider the following two statements: The $*$-subalgebra generated by $1$ and $z$ spans ...
-2
votes
0answers
14 views

functional analysis. Riesz system. [on hold]

I have the following problem: "a) Let $H$ --- Hilbert space, $T : H \to H $ --- unitary operator. For the vector $x_0 \in H$, we assume that $x_j := T^j x_0, j = 0,-1,+ 1,-2,+2 ... $ suppose that ...
0
votes
0answers
28 views

Is there any relation Trace and Boundary?

I understand the trace is sum of diagonal elements of a matrix. Further the boundary I always perceive as a 'end points' of bounded domain. However on the link below: ...
1
vote
2answers
19 views

Does $a_n \in H^1(\mathbb R^n)$ and $b_n \rightharpoonup 0$ in $H^1(\mathbb R^n)$ imply $\langle a_n, b_n \rangle \to 0$?

I have a question mainly in functional analysis. Suppose that $H^1(\mathbb R^n)$ is the standard Sobolev space that we all know. My question is as follows: Does $a_n \in H^1(\mathbb R^n)$, $|a_n| ...