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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25 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.
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18 views

John B.conway chapter 2 sec 4 exe 3 [on hold]

If T∈B_00(H,K) show that T*∈B_00(K,H) and dim(ranT)=dim(ran T*)
0
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8 views

Calculus of variational in 2 dimension with constraints

Let $S$ be a 2D region and its boundary is $\partial S$. $u(x,y)$ is defined in S. The functional is of the following type: $J[u] = \int_S F(x,y,u,u_{x},u_{y}) \mathrm{d}s + \int_{\partial S} ...
0
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0answers
14 views

Classical solution involving semigroups

Let $\{T_{D}(t)\}_{t\ge 0}$ a $C_{0}$-semigroup with generator $A+D$, with $D\in\mathcal{L}(X)$ and let $x_{0}\in \mathcal{D}(A)$. I want to show that ...
4
votes
1answer
25 views

Defining a bounded operator on $l^p$

Let $(c_{jk})_{j,k \in \mathbb{N}} \subset \mathbb{C}$ be such that $a:=\sup_{k \in \mathbb{N}} \sum_{j \in \mathbb{N}}|c_{jk}|<\infty$ and $b:=\sup_{j \in \mathbb{N}} \sum_{k \in ...
2
votes
1answer
18 views

Prob. 14, Sec. 3.8 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: A Hermitian positive semi-definite form

Let $X$ be a complex vector space, and let the map $h \colon X \times X \to \mathbb{C}$ satisfy the following conditions: For all $x, y, z \in X$ and $\alpha \in \mathbb{C}$, (i) $h(x+y, z) = ...
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0answers
21 views

A Hilbert space is isomorphic to its second dual?

How to show that any Hilbert space $H$ is isomorphic to its second dual space $H^{\prime\prime} = (H^\prime)^\prime$? (This is Prob.8, Sec. 3.8 in Erwine Kreyszig's Introductory Functional Analysis ...
0
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0answers
24 views

If $x \in c_0 $ and $\sum_{i=1}^{\infty}|x_n|< \infty $ $ \implies x \in l^p$ for $p \in (1,\infty)$

Hi I was trying to solve my problem sheet in functional analysis. In order to prove something I need to show that If $x \in c_0$ and $\sum_{i=1}^{\infty}|x_n|< \infty $$ \implies x \in l^p$ for $p ...
0
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0answers
7 views

Verification of the Hormander condition [on hold]

please, can someone to tell me, how to show that two vectors in R3 span R3, by using the hormander condition.
2
votes
1answer
30 views

$X\times Y$ is a banach space. What can you say about $X$ and $Y$?

$X\times Y$ denotes the cartesian product of $X$ and $Y$. $X\times Y$ is a Banach space with respect to the norm $$||(x,y)||:= (||x||^p + ||y||^p)^{\frac 1p}.$$ What can I say about $X$ and $Y$ ...
0
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0answers
13 views

the Gelfand and norm topologies are equal on character space

We know that the character space of the Banach algebra $L^{1}(Z)$ is homomorphic to the unit circle $T$, but we can't show that the Gelfand and norm topologies are equal on that.
1
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1answer
29 views

Prob. 7, Sec. 3.8 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: The dual space of a Hilbert space is a Hilbert space.

Here's Prob. 7, Sec. 3.8 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Show that the dual space $H^\prime$ of a Hilbert space $H$ is a Hilbert space with inner product ...
1
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0answers
14 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 = ...
0
votes
1answer
35 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
votes
0answers
10 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
votes
1answer
101 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 ...
8
votes
1answer
63 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 ...
-3
votes
1answer
50 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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1answer
31 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
votes
3answers
38 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 ...
9
votes
6answers
933 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 anyone help me? I don't need examples or the definition - I want to know why and ...
5
votes
2answers
217 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
55 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 ...
0
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0answers
13 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
34 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 ...
0
votes
1answer
24 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
36 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 ...
1
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0answers
15 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
40 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
26 views

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

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
56 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
vote
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
votes
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 ...
1
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2answers
33 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.
1
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0answers
34 views

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

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
votes
0answers
32 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 ...
0
votes
0answers
21 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
votes
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
votes
0answers
33 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. ...
1
vote
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
28 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
72 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
vote
1answer
28 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
37 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
votes
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 ...