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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2
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1answer
50 views

Calculate $\|f\|$ with $f(x) = \int_{-1}^{1}x(t)dt - 2x(0)$

In $C[-1,1]$, consider the norm $\|x\| = \max\{x(t)\mid t \in [-1,1]\}$ and $f(x) = \int_{-1}^{1}x(t)dt - 2x(0)$, $x \in C[-1,1]$. Find $\|f\|$ Hi everybody. I got stuck in this problem. I can ...
1
vote
1answer
40 views

If a set $A$ is disconnected in $(X,d_1)$, then it is disconnected in $(X,d_2)$ for any metric $d_2\geq d_1$

If a set $A$ is disconnected in metric space $(X,d_1)$, then it is disconnected in $(X,d_2)$ for any metric $d_2\geq d_1$ we need to prove or disprove. we think it is true. any open set for $d_1$ is ...
0
votes
1answer
146 views

how to show a countable space is totally disconnected for any metric?

Suppose X is countable. We need to show that for any metric d on X the space (X,d) is totally disconnected. It is true that any subset of a countable set is countable. so, divide the space until its ...
0
votes
2answers
111 views

A question about weighted forward unilateral shift operators

We define $$ B(x_{1}, x_{2},...)=(0, \frac{x_{1}}{2}, \frac{x_{2}}{3},...,x_{n})\in l^{2}(N), $$ How could be shown that that $B$ is a quasinilpotent?
0
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1answer
141 views

Convolution of functions and measures

I need some help with this exercise. I'm not sure how to deal with it: Let $f(x)=e^{-x^2}$, $\mu$ the Lebesgue measure in $[0,1]$ and $\nu$ the Lebesgue measure in $[2,\infty)$. I have to find the ...
0
votes
1answer
102 views

every denting point and strongly exposed point is extreme point

If $X$ be a Banach space and $K$ is a subset of $X$, then I want to prove Every denting point of $K$ is extreme point Every strongly exposed point of $K$ is extreme point $K$ is the closed convex ...
1
vote
2answers
42 views

The space of regular curves deformation retracts onto the space of arclength parameterized curves

Let X denote the space of smooth maps from the circle into R^3 which have no zero derivatives. This is an open submanifold of the Frechet space of smooth maps from the circle into R^3. Let Y denote ...
0
votes
1answer
85 views

about orthogonal complement

Im reading a chapter talking about orthogonal complement of dual space in optimization by vector space. I have captured something confusing as following: I cannot understand some parts of the proof ...
2
votes
0answers
39 views

Riemann integrability in not sequentially complete LCS?

For $E$ any Hausdorff locally convex space, I have been wondering whether Riemann integrability of all continuous functions $f:[\,0,1\,]\to E$ implies that $E$ be sequentially complete. For example, ...
3
votes
1answer
63 views

Radon integrals and Radon charges

I'm reading chapter 6 (measure theory) of Pedersen's book Analysis now and I'm a bit puzzled in his passage from Radon integrals to Radon charges. The book in 6.1.2 defines a Radon integral to be a ...
0
votes
1answer
76 views

Tempered fundamental solutions

According to the Malgrange–Ehrenpreis theorem every nontrivial linear constant coefficient PDO $P(\partial)$ admits a fundamental solution $E\in\mathscr{D}'$; I wonder whether $P(\partial)$ admits a ...
2
votes
1answer
124 views

Show that $\lim_n \|\partial^s (f_n - g_n)\|_p = 0$ (no homework…)

the setting is as follows: Let $\Omega \subset \mathbb{R}^m$ be open and consider some $L^p(\Omega)$ which I will shortly write as just $L^p$ from now on. Furthermore let (for some $k \in ...
0
votes
1answer
64 views

why for every $ f\in C(\sigma(x))$ we have $ \Phi (f(x))= f(\Phi(x))$?

In a book about $ C^* $-algebra, in the section of continuous functional calculus says that: Suppose $ x $ is a normal element of $ C^*$-algebra $ A $, then the continuous functional calculus ...
5
votes
1answer
91 views

About measurability of operators

I'm triyng without success, to find some examples of functions that: $\bullet$Are WOT-measurable, but not SOT-measurable. $\bullet$Are SOT-measurable, but not $||\cdot||$-measurable. I give the ...
0
votes
1answer
65 views

Prove x(t) is bounded given a integral inequality

I want to answer the following question: $x=x(t)$ is defined and continuous on $[0,T)$ and satisfies an integral inequality $$1 \leq x(t) \leq A_1 + A_2\int_0^t x(s)\big(1+\log x(s)\big) ds$$ for ...
0
votes
1answer
192 views

Quotient norm and actual norm

I have a question about the proof that $X\backslash U$ is a Banachspace if $X$ is one and $U$ is closed. In my book it is said, that for $x_k \in X$ and a series $\sum_{k=1}^{\infty}||[x_k]||< ...
0
votes
1answer
118 views

about dual space

Im reading a chapter talking about dual space in optimization by vector space. I have captured something confusing as follow: I have been confused by two sentenses that one is " The mapping φ:X -> ...
0
votes
1answer
125 views

Show that orthogonal complement is trivial

I have this subspace of $C[-1,1]$ with inner product $\langle f,g\rangle = \int_{-1}^1f(x)\cdot \bar g(x)\,dx$: $$ E=\left\{f : \int_{-1}^0f=\int_{0}^1f\right\} $$ need to prove that $E^\bot=\{0\}$
3
votes
1answer
124 views

Equicontinuity and Uniform Boundedness

If we have a sequence of smooth functions $\{f_{n}\}_{n}$ where $f_{n}: U \rightarrow \mathbb{R}$, where $U \subset \mathbb{R}^{n}$. We are given the following two results: For $x \in U$ we have ...
4
votes
2answers
222 views

Using the Extension Operator Theorem for Sobolev Spaces

I want to know if certain conditions hold after applying the Sobolev Extension Theorem: Assume $U$ is a bounded open subset of $\mathbb{R}^{n}$ and $\partial U$ is $C^{1}$. Suppose $1 \leq p < n$. ...
1
vote
0answers
46 views

A question about tensor product of algebras of compact operators. [duplicate]

Let $\cal{H}$ be a separable Hilbert space and $\cal{K(\cal{H})}$ the algebra of compact operators acting on $\cal{H}$. Then $$\cal{K(\cal{H})}\otimes\cal{K}(\cal{H})\cong\cal{K}(\cal{H}\otimes H).$$ ...
2
votes
3answers
121 views

topology on $\hom_{\mathbb C}(V,W)$

Let's $V,W$ be finite-dimensional complex vector spaces. How to define topology on $\hom_{\mathbb C}(V,W)$? As I see we can define metric in this $(\dim_{\mathbb C}V\times\dim_{\mathbb ...
2
votes
0answers
79 views

show that linear functional is unbounded

Let $F:C^{1}[0,1]\to \mathbb{C}$ be equipped with the supremum norm $||.||_{\infty}$, $F(f)=f^{\prime}(1)$. I am trying to show that $F$ is unbounded. Here is my idea. I take a sequence ...
1
vote
1answer
95 views

Show existence of a continuous function with certain properties

Let $X$ be a compact Hausdorff space and $C(X)$ the commutative algebra of continuous complex-valued functions endowed with the maximum-norm. Let $J\subset C(X)$ be an ideal and $g\in J$. Let ...
2
votes
1answer
125 views

Is this a legitimate way to make a non-measurable set?

I thought up a way of creating a non-measurable set, and I'd appreciate some input. Thanks! Let $G$ be a countable group acting on an uncountable measure space $X: \mu(X)<\infty, |X|=\aleph$. ...
3
votes
0answers
143 views

Prokhorov theorem in locally compact Hausdorff space?

Prokhorov theorem gives a compactness condition in the space of probability measures on a Polish space. I am wondering whether we have similar conditions for probability measures on more general ...
1
vote
1answer
162 views

Banach algebra problem: $\left\|e^{ta}\right\|\leq Me^{-\omega t}$

Let $A$ be a unital algebra, and $a\in A$. Assume that $\sigma(a)\subset \{\lambda\in \mathbb{C}: Re\lambda < 0\}$. Show there exists $M,\omega >0$ such that $$\left\|e^{ta}\right\|\leq ...
2
votes
1answer
100 views

Measure-valued maps equal a.e. if their integrals equal over any set

Let $X,Y$ be two standard Borel spaces and let $p,q:X\to\mathcal P(Y)$ be two stochastic kernels, which can be alternatively seen as measure-valued maps. Suppose that for some measure $\mu\in \mathcal ...
1
vote
1answer
156 views

Convolution of distributions is not associative

I need some help with this exercise: It proposes to show that convolution of distributions is not associative: If $T=T_1$ (distribution given by f=1), $S=\delta'$, and $R=T_H$ (we denote as $H$ the ...
1
vote
1answer
41 views

Spectral characterization of induced operator norm

Consider $\mathbb{R}^n$ with the $l^1$ norm and the induced operator norm $\| \cdot \|$ on linear maps $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$. Can $\|T\|$ be characterized somehow by the spectrum ...
1
vote
1answer
69 views

connected neighborhood of $0$ must be in unit ball

In topological vector space $C^{n}$, let $B$ be the open unit ball and $S$ be the unit sphere. Suppose $E$ is a connected neighborhood of $0$ disjoint from $S$. Prove that $E \subset B$ ...
0
votes
1answer
158 views

Closed mapping theorem

I have been trying to prove the following: Assume that $X$,$Y$ are two normed spaces. Let $T$ be a closed mapping $T: X \to Y$ show that the image $A$ of a compact subset $C$ in $X$ is closed. Show ...
2
votes
1answer
214 views

The Arzelà–Ascoli theorem fails on a half-open interval

Can we find an example: (1) $\lbrace f_n \rbrace_n$ is a family of real-valued functions defined on $[0,1)$ such that this family is uniformly bounded and equicontinuous, $f_n(0)=0$; ~~~ Uniformly ...
0
votes
1answer
27 views

Is the collection of hyperplane separating vectors Borel-measurable?

Let $C\subseteq\mathbb R^d$ be non-empty, convex, such that $0\notin C$. Let $$H=\{\alpha\in\mathbb R^d\mid\alpha\cdot c\geq 0 \text{ for all } c\in C\text{ and }\alpha\cdot c_0>0 \text{ for some } ...
1
vote
1answer
51 views

Norm of a linear mapping, please check if what I have done is right

please check if what I have done is right. $C[0,1]=$ continuous functions in $[0,1]$ considering $\|g\|=\max_t|g(t)|$ $$X=\langle t^2,1 \rangle $$ the subspace of $C[0,1]$ generated by $t^2$ and $1$ ...
0
votes
1answer
234 views

The gradient of the standard mollifier

Please check my proof for the following result: I want to prove a result for $D\eta_{\epsilon}$ the gradient of the standard mollifier $\eta$. The function $\eta$ is defined as follows: Let ...
2
votes
1answer
55 views

When can we interchange Fourier transform and countable sum?

When does $\mathcal{F}\left ( \sum_{n=1}^{\infty} f_n (x)\right ) = \sum_{n=1}^{\infty} \mathcal{F}(f_n(x))$ where $\mathcal{F}$ the Fourier transform operator.
0
votes
1answer
31 views

$W = \{x\in l_0 : <x,a>=0\}$ where $a=(1,\dfrac12,\dfrac13,…)$ and $l_0$ is sequences with finitely many non-zero terms. Show $W$ is separable

Consider the inner product space $l_0$ consisting of all infinite sequences of complex numbers with only finitely many non-zero terms, with the inner product of $l^2$ (space of square summable ...
1
vote
0answers
59 views

Eigenfunction Expansion for Simple Nonhomogenous PDE

How do I go about finding an eigenfunction expansion for the following equation: $$ u'' = f(x)$$ where: $$ u'(0) = \alpha \quad u'(1) = \beta$$ What about the case when $f(x) = C$ a constant? Edit: ...
0
votes
1answer
28 views

Inequality with partial integration in one dimension

Is it possible to prove $ \| u \|_{L^2(0,1)} \leq \| u' \|_{L^2(0,1)} $ for $u \in C^1([0,1])$ with $u(0)=0$ by using partial integration?
1
vote
0answers
24 views

Solve an integral equation in an Hilbert space

Let $V_n\subseteq [H(div;\Omega)]^{2\times 2}$ and $Q_n\subseteq [L^2(\Omega)]^2$ two finte dimentional spaces such that $div(H_n)\subseteq Q_n$. Suppose that $u\in Q_n$ is well known. I must solve ...
0
votes
1answer
30 views

determine the maximal open subset $G\subset\mathbb{C}^2$ such that the series $f$ converges in $\mathcal{O}(G)$ in the topology of Frechet space.

Consider $f(z_1,z_2)=\sum\limits_{j=0}^\infty(z_1+z_2)^j$,determine the maximal open subset $G\subset\mathbb{C}^2$ such that the series $f$ converges in $\mathcal{O}(G)$ in the topology of Frechet ...
0
votes
1answer
48 views

$A = \{e_i:i\in \mathbb{N}\}$ where $e_i = (0,…0,1,0,0,…)$. Show that $\overline{span(A)}=c_0$, set of infinite sequences that converges to 0

Let $A = \{e_i:i\in \mathbb{N}\}$ where $e_i = (0,...0,1,0,0,...)$. It has $1$ in the $j^{th}$ entry. Define $span(A)$ as the set consisting of all finite linear combinations of elements in $A$. ...
0
votes
1answer
42 views

Is $u \mapsto u'(0)$ continuous on $C^1([0,1])$ with the $C^0$-norm?

I need some help with this exercise: Is the functional $D$ defined by $u\mapsto u'(0)$ for $u\in C^1([0,1])$ linear and continuous? First of all: $C^1([0,1])$ is the space of all functions ...
3
votes
2answers
104 views

Disintegration-like theorem

$\def\b{\mathcal B}\def\p{\mathcal P}\def\d{\mathrm d}$ Let $X$ be a (standard) Borel space: a topological space isomorphic to a Borel set of a complete separable metric space. Denote by $\b(X)$ the ...
4
votes
1answer
109 views

Two question on a lemma about C*-algebra

I am reading Lin Hua xin's book "An introduction to the classification of amenable C*-algebras" and i am confused with the lemma 1.7.12 in this book. Lemma 1.7.12 Let $A$ be a C*-algebra and $f\in ...
6
votes
1answer
141 views

Can $f*g = f+g$ for $f$ and $g$ compactly supported?

Let $f$ and $g$ be continuous, compactly-supported functions $\mathbb{R} \to \mathbb{C}$. Can it happen that $f*g = f+g$? Here, $f*g$ denotes the convolution $$(f*g)(s) = \int_\mathbb{R} f(t) g(s-t) ...
2
votes
1answer
480 views

Showing when Young's Inequality is in fact equality.

The ellipses is where I'm stuck. I don't think a simple algebraic manipulation will work here.
2
votes
1answer
272 views

Application of Uniform Boundedness Theorem to prove an equivalence involving sequences.

After state and prove the Uniform Boundedness Theorem, the Kreyszig Functional Analysis book presents the following problem: I'm trying to solve it but I need help to finish it. What I have done ...
3
votes
1answer
186 views

Is there a formula for the Haar measure on a product of groups?

Let $ (G_{n})_{n \in \mathbb{N}} $ be a sequence of locally compact topological groups with a corresponding sequence $ (\mu_{n})_{n \in \mathbb{N}} $ of Haar measures. Is there a way to construct a ...