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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5 views

linear operator vs. quasilinear operator

Let $\mathcal{T}$ be an operator defined on a linear space of complex-valued measurable functions on a measure space $(X,\mu)$ and taking values in the set of all complex-valued finite almost ...
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0answers
9 views

Kolmogorov's superposition theorem for non-continuous functions

I'm trying to think about Kolmogorov's superposition theorem. This theorem states that, for each $n ≥ 2$ there exist continuous functions $ϕ_q : [0, 1] → R, q = 0, ..., 2n$ and constants $λ_p ∈ R, p = ...
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1answer
30 views

Category theorem

I don't have a mathematician background (I am engineer) I understand some concepts but still very abstract for me and I have to show the following: 1.- Of what category is the set of all rational ...
0
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0answers
6 views

Comparison between interpolation and Tikhonov regularization.

Interpolation is defined as finding a value of a function between two points and one can think of Tikhonov regularization as to estimate a suitable function under certain condition. Can we think ...
3
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0answers
64 views

Existence of a minimizer for $\int_0^1|P(t)|\,{\rm d}t$.

Let $m > 0$ be a fixed integer. Show that among all the polynomials $P \in \Bbb C[X]$ with degree $\leq m$ and with $P(0)=1$, there is one that makes minimum the value $\int_0^1|P(t)|\,{\rm ...
7
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1answer
55 views

Proving that every vector space has a norm.

I am trying to prove that every vector space $X$ has a norm. I have some silly questions, but it's better to ask them now instead of later. I think I'm having a bit of trouble getting intuition about ...
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1answer
44 views

John b.Conway chapter $2$ section $2$ exercise $4$ [on hold]

Show that an idempotent is compact if and only if it has finite rank.
4
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0answers
25 views

$\overline{\mathrm{Im} (T^*T)} = \overline{\mathrm{Im} T^*}$

I need to prove that in a Hilbert space, $\overline{\mathrm{Im}(T^*T)} = \overline{\mathrm{Im}T^*}$. I have already shown that $\ker (T^*) = (\mathrm{Im} T)^\perp$ and have so far concluded that ...
2
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3answers
35 views

Name of Jordan Canonical Form in infinite dimensions?

I tend to think of Jordan canonical form as the generalized spectrum theorem. I read it as saying, every matrix cannot be diagonalized, but they can be "jordanized". In functional, I've seen the ...
6
votes
7answers
494 views

What is the idea behind a projection operator?what does it do?

I know what a projection operator is, but I am unable to explain it in words without using mathematical symbols. Can any one help me? I don't need the examples or the def I need to know why and how ...
5
votes
2answers
215 views

Fourier Transform: Understanding change of basis property with ideas from linear algebra

The notion of Fourier transform was always a little bit mysterious to me and recently I was introduced to functional analysis. I am a beginner in this field but still I am almost seeing that the ...
3
votes
0answers
49 views

The Mountain Pass theorem

I cam across the Mountain Pass Theorem, mentioned for example at http://en.wikipedia.org/wiki/Mountain_pass_theorem. In (very) loose terms, it somewhat reminds me of Rolle's theorem. Trying to ...
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0answers
11 views

Geometry Meaning of Helly's Theorem

I am studying about reflexive space in the book Funtional Analysis, Sobolev Spaces, and Partial Differential Equations Haim Brezis. Could someone help me the clarify the geometry meaning of Helly's ...
1
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1answer
30 views

Theorem 3.8-1 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: Do we really need the completeness of the space?

Here's Theorem 3.8-1 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Every bounded linear functional $f$ on a Hilbert space $H$ can be represented in terms of the inner ...
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1answer
19 views

a question on the extension of an operator?.

It is known that $C_0^{\infty}(\Omega)$ is dense in $W_0^{1,p}(\Omega)$ where $\Omega$ is a bounded domain of $\mathbb{R}^n$. Let $T:C_0^{\infty}(\Omega)\rightarrow\mathbb{R}$ be a continuous linear ...
1
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1answer
33 views

Spectral theory for $f\mapsto f\circ g$

Consider the Banach space $B = C([0,1] \to \mathbb R)$ of continuous functions from $[0,1] \to \mathbb R$ with the supremum norm. Let $g$ be a continuous function $g:[0,1] \to [0,1]$. Then one can ...
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0answers
14 views

Convergence in distribution of distributions $p_n$ implies convergence in distribution of $s_n$?

Question Setup Suppose $p_n(x,y)$ is a sequence of probability densities on $\mathbb R^2$ and $q_n(x)$ is a sequence of densities on $\mathbb R$ such that \begin{align*} \int b(x,y) \ p_n(x,y) \ dx ...
2
votes
2answers
39 views

multiplying by a $C^\infty$ function

If $f \in C^\infty$ and $g$ is a real valued function can we say anything about their product? In particular is $fg \in C^\infty$ or maybe if we stipulate $g$ has compact support can we make the ...
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0answers
24 views

Show that the functionals $\{f_1, f_2, f_3 \}$ form the base of the real space of polynomials $P_2 (\mathbb{R})$.

In the area of real polynomial $P_2(\mathbb{R})$ are given functional ! $$f_i(p) := 6 \int_0^i p(t) dt; \space \space i = 1,2,3$$ $$(p,q) := \int_{-1}^1 p(t)q(t) dt$$ Which polynomial by Riesz ...
1
vote
2answers
53 views

Let $f:(a,b)\rightarrow \mathbb{R}$ be non-decreasing i.e. $f(x_1)\leq f(x_2)$ 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. $f(x_1)\leq f(x_2)$ and let $c \in (a,b)$. Show that $\lim_{x \ \rightarrow ...
1
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1answer
18 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 ...
0
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0answers
32 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
32 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
30 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 ...
2
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0answers
31 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
19 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 ...
0
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0answers
52 views

A function continuous on rational points and discontinuous on irrational points [duplicate]

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 ...
0
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0answers
31 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
31 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
votes
1answer
29 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 ...
1
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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?
1
vote
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 ...
9
votes
2answers
66 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
vote
1answer
15 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 ...
1
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1answer
27 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
vote
1answer
36 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 ...
0
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1answer
31 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 ...
-5
votes
0answers
24 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, ...
-1
votes
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 ...
0
votes
1answer
17 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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votes
1answer
31 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.
1
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0answers
20 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
52 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
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0answers
40 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 $X$. ...
1
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0answers
17 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
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 ...
1
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2answers
46 views

Clarification of notation $\|fw\|$

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
21 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
177 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
18 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 ...