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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7 views

Let $f:(a,b)\rightarrow \mathbb{R}$ be non-decreasing i.e. $f(x_1)\leq g(x_2)$ and let $c \in (a,b)$. Show that limits from both sides exist.

I would like to ask you a question about the following question. Let $f:(a,b)\rightarrow \mathbb{R}$ be non-decreasing i.e. $f(x_1)\leq g(x_2)$ and let $c \in (a,b)$. Show that $\lim_{x \ \rightarrow ...
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11 views

Compute an integral with Cauchy's residue theorem

Good evening everyone, I want to know if my result is correct. So: I have to compute the following integral: $$\int_\gamma \frac{ze^{\pi z}}{z^2+1}dz,$$ while ...
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0answers
23 views

Weak and weak$^*$ topologies

I have some confusion about weak and weak* topologies. What I have understood is as bellow. Let $X$ be a normed linear space and let $X^*$ be its topological dual. Then the coarsest Hausedorff ...
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2answers
26 views

Inverse of $I +T^*T$

I am trying to show that the inverse of the operator $I +T^*T$ exists. What I have been trying to do is trial and error taking inverses of $T$ and $T^*$ but to no avail.
2
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0answers
22 views

Theorem 3.6-2 in Erwine Kreyszig's “Introductory Functional Analysis with Applications:” Does the converse hold if the space is not complete?

First, a definition: Let $X$ be a normed space. A subset $M (\neq \emptyset) \subset X$ is said to be total in $X$ if the span of $M$ is dense in $X$. Now theorem 3.6-2 in Kreyszig states the ...
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0answers
20 views

upper bound of a differential equation solution

Let $A(t)$ be a bounded singular values matrix that is function of time, and $f(t)$ and $L^\infty$ function of time. And consider the ODE $$ \dot x = A(t) x + f(t) $$ How we can describe qualitatively ...
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0answers
13 views

why the set of continuous complex-valued functions on an open set of $R^n$ is not normable?

why the set of continuous complex-valued functions on an open set of $R^n$ is not normable? I am trying to follow example 1.44 in Rudin's Functional Analysis book, to show that if: $\Omega$ is ...
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0answers
40 views

A function continuous on rational points and discontinuous on irrational points

How to find function $f : \Bbb R \to \Bbb R$ such that $f$ is continuous on the rational numbers and discontinuous at irrational numbers? I was told to use the Baire Theorem to show that the set of ...
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0answers
23 views

upper bound of an $L^\infty$ function's derivative

Consider a function $u:\mathbb{R} \longrightarrow \mathbb{R}^n$ that is essentially bounded, i.e., $u \in L^\infty$. There is an upper bound of its derivative? I think there is not allways ( i.g. ...
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3answers
26 views

$A$ and $B$ are bounded linear operators from the normed linear space $X$ to itself. If $AB$ is invertible are $A$ and $B$ invertible?

I think I understand the proof for square matrices, such that $(AB)^{-1}=B^{-1}A^{-1}$, but I'm not sure if I can just say the same for the bounded linear operators A and B. Does the existence of ...
2
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1answer
25 views

Existence of a global limit in $L^1([-N,N])$ for each $N\in \mathbb{N}$

Let $(f_n)_n$ be sequence of functions $f_n\in L^1_{loc}(\mathbb{R})$ such that for each $N\in \mathbb{N}$, $(f_n)_n$ is a Cauchy sequence in $L^1([-N,N])$. Then for each $N$, $(f_n)_n$ converges to a ...
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0answers
25 views

the set of continuous complex-valued functions on an open set of $\Bbb R^n$ is not normable

why the set of continuous complex-valued functions on an open set of $R^n$ is not normable?
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2answers
26 views

Interior, closure, isolated points and boundary of a set of a normed vector space

Let $X =(\mathbb{R}^2,||(x_1,x_2)|| := |x_1| +|x_2|)$ be a normed vector space. Find the interior, closure,Isolated points, and boundary of $Y =\{(x, \frac{1}{n})~|~ x\in \mathbb{R} \wedge n\in ...
7
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1answer
29 views

A question about the vector space spanned by shifts of a given function

Let $E$ be the space of continuous real functions and $f\in E$ Let $T_t$ denote the shift operator: $T_t(f)(x)=f(t+x)$ Let $T(f)$ be the linear span of the set $\{T_t(f) \;|\; t\in ...
1
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1answer
11 views

Biorthogonal complement of subspace of subspace.

I'm taking a course on Banach and Hilbert spaces. The teacher who guides the exercise sessions is often a bit fast, so only when revising my notes at home I realize I do not fully understand them. We ...
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1answer
25 views

$l^p$ space not having inner product

I know that $l^2$ space is a Hilbert space. But for other $l^p$ spaces, where $p\geq1$, I have to show that they do not satisfy the parallelogram equality. But, I can't find appropriate sequences ...
1
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1answer
33 views

A bounded sequence in a Banach space

Let $X$ be a Banach space and $\langle x_n\rangle $ be a sequence in $X$. If ( $f(x_n)$ ) is a bounded sequence for any bounded linear functional $f$ on $X$, then ( $x_n$ ) is a bounded sequence in ...
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1answer
29 views

Can't work out if this proof is sound or not. Any ideas?

Let $V$ be a normed space over some field $\mathbb K$. I proved that $$ \overline{B_r(a)} = \{v \in V \mid \|v-a\| \le r \}$$ $\subseteq $ was easy but for the $\supseteq$ direction I am really not ...
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0answers
23 views

show or find the adjoint operator of an operator [on hold]

Show that the adjoint operator of the zero operator is zero., help meee and showed the identity of the operator adjoint operator but this , although it does not give me the idea. I have this idea, ...
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0answers
14 views

A space of complex convergent sequeces [duplicate]

I am just stuck at an exercise problem regarding the space of all complex convergent sequences. The norm on this space is given by the supremum of each sequence. If f is a bounded linear functional ...
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1answer
14 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, ...
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1answer
30 views

Show that the adjoint operator of the zero operator is zero [on hold]

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.
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0answers
18 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
49 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 ...
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0answers
33 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 $X$. ...
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0answers
15 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 ...
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1answer
10 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 ...
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2answers
36 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
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2answers
20 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
176 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
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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 ...
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0answers
16 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.
2
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0answers
26 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
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1answer
29 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
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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
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1answer
38 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 ...
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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
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2answers
33 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 ...
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0answers
24 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 ...
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0answers
14 views

Hilbert space and Orthogonal complement [on hold]

Let $M$be a linear subspace of Hilbert space $H$. Show that $$cl M = H \Leftrightarrow M^{\perp} = \left\{ 0 \right\}$$
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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 ...
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0answers
44 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 ...
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0answers
26 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 ...
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0answers
14 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
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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 $\rho: X \to \mathbb{R}^+_0$ ...
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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
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1answer
32 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
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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 = ...
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1answer
32 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
80 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 ...