Tagged Questions

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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13 views

Nonlinear continuous and unbounded operator [duplicate]

Let $X$ be an infinite-dimensional Banach space, and let $B=\{x\in X: \|x\|\leq 1\}$ be its closed unit ball. Does there exist a continuous mapping $F: X\to X$ such that the set $F(B)=\{F(x): x\in ...
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1answer
22 views

Weak convergence in Banach spaces

In Rudin's book <> Page 66, it says "If $X$ is a infinite dimensional topology vector space, then $X$ under the weak topology is not locally bounded" . Hence I think the topology of any ...
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1answer
15 views

Generalised derivative and derivative of functions of bounded variation

Let $f:\mathbb{R}\to\mathbb{C}$ be a function Lebesgue-integrable on any finite interval and let $K$ be the space of infinitely differentiable equal to 0 outside a given finite interval. Be the ...
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2answers
18 views

Finding simplest function to distinguish 2 sets

I wish to find a function that distinguishes $2$ sets. I have m data values in form of n-tuples out of which k are supposed to be mapped to a value less than $0$ and other m-k are supposed to be ...
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1answer
12 views

Is the monotone convergence theorem bidirectional?

Say I have $(f_n)$ with $f_1 \le f_2 \le ...$ and I know that $\lim_n\int f_n<\infty$ exists, does that imply $f_n$ converges a.e.? Most formulations I have seen of the monotone convergence ...
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26 views

James $\ell_1$-theorem: problem in the proof

I am struggling with the very last estimate in the proof of James' $\ell_1$-theorem. (Please see below an excerpt from Albiac and Kalton's fantastic book Topics in Banach space theory.) I don't see ...
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16 views

Power series for functionals and notation for functionals

I am trying to learn some functional analysis/calculus of variations, mainly for being able to perform functional derivatives on simple functionals found in physics (therefore I will not be too ...
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27 views

Alternative proof of "Every linear mapping on a finite dimensional space is continuous”

Here is my question: Suppose that $T:X\to Y$ is linear, where $X$ and $Y$ are normed linear spaces, and $X$ is finite dimensional. Define $\|\cdot\|_\beta$ on $X$ by ...
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26 views

$X$ is the vector space $C[0,1]$ with the norm $\|f\|_1=\int_0^1|f(t)|dt$, and $M=\{f\in X:f(0)=0\}$, show that $M$ is not closed.

Here is my question: Let $X$ be the vector space $C[0,1]$ with the norm $\|f\|_1=\int_0^1|f(t)|dt$. Let $$M=\{f\in X:f(0)=0\}$$ Show that $M$ is not closed. Show that the “quotient norm" ...
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1answer
28 views

$X=C[0,1]$ is a Banach space, $M=\{f\in X: f(0)=0\}$, prove $M$ is closed, find explicit formula for the quotient norm, and find an isomorphism.

Here is my question: Let $X$ be a Banach space $C[0,1]$ with the supremum norm. Let $M=\{f\in X: f(0)=0\}$. Show that $M$ is closed. Find an explicit formula for the quotient norm $\|[f]\|$ for ...
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19 views

Show that $\lbrace S_n x \rbrace$ converges for a particular recursively-defined sequence of operators $S_n$

H is a Hilbert space, M is a self-adjoint bounded linear operator on H with $M \leq I$, and $S_0 = 0$; $S_{n+1} = (1/2)(M + S^2_n)$ for $n = 0, 1, 2, ...$. For all n, both $S_n$ and $S_n - S_{n-1}$ ...
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2answers
31 views

If $\sum_{n=1}^\infty \|x_n\|\lt\infty$, , then $\lim_{k\to\infty}\sum_{n=1}^k x_n$ exists

Here is my question: Let $X$ be a Banach space with norm $\|·\|$. Prove that, for any sequence $\{x_n\}$ in $X$, if $\sum_{n=1}^\infty \|x_n\|\lt\infty$, then $\lim_{k\to\infty}\sum_{n=1}^k x_n$ ...
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0answers
13 views

About a critical nonlinearity

I am reading the following book: Concentration Compactness functional-analytic grounds and applications (By Kyril Tintarev and Karl-Heinz Fieseler) and i am stuck in the following problem (Problem 5.2 ...
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1answer
30 views

Generalised derivative of Cantor staircase

If we consider the Cantor staircase function, let us say $f:[0,1]\to\mathbb{R}$, as a distribution, I was wondering whether there is an explicit way to express its generalised derivative as a ...
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1answer
20 views

Verifying a bound on the norm of an operator in $l_2$.

The problem: Define $L: l_2 \rightarrow l_2$ by $L(x_1, x_2, ...) = (y_1, y_2, ...)$, where $y_n = (x_1 + x_2 + ... + x_n)/n^2$. Show that $||L|| \leq (\sum_{n=1}^\infty 1/n^2)^{1/2}$. My proof: ...
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2answers
18 views

Equivalance of norms

Let $X$ be the vector space of all real valued functions defined on $[0,1]$ having continuous first-order derivatives. How to show that the following norms are equivalent: $\|f\|_1 = |f(0)| + ...
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1answer
11 views

How can I complete my proof: Sobolev space W^(1,p) is complete? Using Convergence theorem

I'm trying to prove that W^(1,k) (R) is complete. The steps i Had so far: let {fn} be a cauchy sequence in W^(1,k). therefore {fn} and {dfn} are cauchy sequences in L^p(R), and therefore converge ...
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0answers
26 views

How can i prove that this function doesn't have a second weak derivative?

I'm trying to determine what weak derivatives the function $$ f(x)=\begin{cases} x&\mbox{if }0<x<1,\\ 1&\mbox{if }1\leq x<2, \end{cases} $$ has. I already managed to prove that it ...
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3answers
27 views

Norm of a functional

I'm facing the problem of calculating the norm of the following functional: $\displaystyle \phi : L_p([0,1]) \rightarrow \mathbb{R}, ~~ \phi (f) = \int\limits_{0}^{1} e^x f(x) dx $ I have no idea ...
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0answers
20 views

Restriction of a spectral measure

Let $x$ be a self-adjoint operator on $H$. By spectral theorem, there is a spectral measure $\mu$ correspondence to $*-$ homomorphism $\pi:C(\sigma(x)) \to B(H)$ such that $x=\int_{-||x||}^{||x||} ...
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1answer
35 views

Spectral Measures: Domain Criterion

Given a topological space $\Omega$ and a Hilbert space $\mathcal{H}$. Let $\mathcal{B}(\Omega)$ be its Borel algebra and $\mathcal{B}(\mathcal{H})$ its bounded operators. Moreover, given a spectral ...
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26 views

Convergence uniformly.

Let $\Omega$ a domain bounded and supoose that $u^{\epsilon}(x)\rightarrow u(x)$ uniformly. How do I show that $$ \int_{\Omega}|\nabla u^{\epsilon}|^2\rightarrow\int_{\Omega}|\nabla u|^2? $$
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2answers
38 views

Hilbert space: product and tensor product space

Let $H_1$ and $H_2$ be Hilbert spaces, then I would intuitively define the inner product on $H_1 \times H_2$ by $\langle (x_1,x_2),(y_1,y_2)\rangle = \langle x_1,y_1 \rangle + \langle x_2,y_2 ...
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1answer
50 views

How to decide completeness of $\ell^\infty$?

Let $\ell^\infty$ denote the set of all bounded sequences $x \colon = (\xi_j)_{j=1}^\infty$, $y \colon= (\eta_j)_{j=1}^\infty$ of complex numbers with the metric $d$ defined as follows: $$ d(x,y) ...
3
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1answer
25 views

Correspondence between bounded sesquilinear forms and bounded linear operators

Let $H,K$ are Hilbert spaces, I want to show there is an isometric linear correspondence between bounded sesquilinear forms $S(H,K)$ and bounded linear operators $B(H,K)$. ( $\Phi: B(H,K)\to S(H,K)$ ...
3
votes
1answer
22 views

Positivelinear operator on $L^p$-spaces

Suppose $1<p<\infty$. A linear operator $T \colon L^p(\Omega)\to L^p(\Omega)$ is positive if $f \geq 0$ imply $T(f)\geq 0$ (where $\Omega$ is a measure space). 1) Does there exist a positive ...
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1answer
34 views

The topology on $C^\infty_c(\mathbb{R}^d)$ used for “distributions of compact support”

On the one hand, Eskin's book on PDEs tells me that I should be content to think of this topology as one "described" (not fully, and it's not even clear it's a topology) by the convergence of ...
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1answer
25 views

Convergence of a sequence of linearly independent vectors in normed space

In an infinite dimensional normed vector space is it possible to find a sequence ${v_n}$ of linearly independent vector (so the sequence is a set of linearly independent vectors) each has norm 1 such ...
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1answer
16 views

A set of differential forms, uniformly bounded with their Laplacians, is precompact in $L^2$.

Let $M$ be a compact Riemannian manifold and let $\Delta$ be a Hodge Laplacian on $k$-forms. How to show that the if the set $\{u_\alpha\} \subset C^2(M,\Lambda^k)$ of $C^2$ $k$-forms is uniformly ...
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3answers
68 views

How to prove that $L^p [0,1]$ isn't induced by an inner product? for $p\neq 2$

I'd like to know how could i prove that $L^p [0,1]$ isn't induced by an inner product? (For $p\neq 2$, including $p=\inf$). It is clear to me that i would need to find two functions $f$, $g$ in $L^p$ ...
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1answer
31 views

Spectral Measures: Square Root Lemma

Given a Hilbert space $\mathcal{H}$. Consider a densely defined closed operator $A:\mathcal{D}(A)\to\mathcal{H}$. This gives rise to operators: $$A^*A:\mathcal{D}(A^*A)\to\mathcal{H}$$ ...
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1answer
25 views

Every metric space can be isometrically embedded in a Banach space, so that it's a linearly independent set

I'm trying to prove the following: Every metric space can be isometrically embedded in a Banach space, so that it's a linearly independent set. I came up with the following idea: Let $ (X,d) $ be a ...
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1answer
11 views

product of spaces of bounded linear operators

Let $E$ be a normed space. Let $(F_i)_{i \in I}$ be a family of normed spaces. Show that $\prod_{i \in I}{\mathcal{L}(E, F_i)}$ and $\mathcal{L}(E,\prod_{i \in I}F_i)$ are isometrically isomorphic. ...
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0answers
22 views

Sobolev norm inequality.

I would like to prove or to disprove the following statement. Let $u$ and $v$ be functions in $H^{s}(S^1)$, the for every $s'\leq s$ $$\|uv\|_s\leq (\|u\|_{s}\|v\|_{s'}+\|v\|_{s}\|u\|_{s'}).$$ I ...
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1answer
13 views

annihilator subspace of normed space

Let $E_0$ be a subpace of the normed space $E$. Let $E_0^a = \{f \in E' : f(x)=0 \forall x \in E_0 \}$ $(E'=\mathcal{L}(E,\mathbb{K})$, where $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C})$. Show that ...
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2answers
26 views

Estimate in Sobolev Spaces

Let $u\in H^0(U)\cap H^1_0(U)$ and $v_k\in C^\infty_c$(U) such that $v_k\rightarrow u$ in $H^1_0(U)$ and $w_k\in C^\infty (U)$ such that $w_k \rightarrow u $ in $H^2(U)$. I want to show that $ \int_U ...
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2answers
32 views

Dual space of a finite dimensional normed space

My lecturer gave us this result today in class, but he didn't give a proof, he said we can prove it ourselves, only I'm really struggling to see how to do it. Let $E$ be a normed space with dual ...
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1answer
16 views

How to create distribution function from sketch?

I'm playing with image manipulation based on various mathematical algorighms (such as edge detection). I'm also changing the colors in various ways just to see what comes out of it. Regarding this, ...
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1answer
31 views

Practical convergence in $C^{\infty}_c$

Let $C^{\infty}_c$ be the space of $C^{\infty}$ functions with compact support in $\mathbb{R}$ with the usual topology derived by the convergence in infinity norm in every $C^{k}_c$. I would like to ...
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1answer
17 views

$C([a,b] \times [c,d],X)$ compared to $C([a,b],C([c,d],X))$ and $C([c,d],C([a,b],X))$

Let $C(Y,X)$ be the space of continuous functions from $Y$ to $X$ together with the supremums norm. Here $Y$ is a compact space and $X$ a metric space. Let $a,b,c,d \in \mathbb R$ be finite, with ...
3
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1answer
37 views

Different norm on $\ell_p$-space and Hilbert space

We define $\ell_p=\{(x_n)_{n\in{\mathbb{N}}}\in\mathbb{C}^\infty:\sum_n{|x_n|^p}<\infty\}$. With the usual usual norm $||.||_p$ this becomes a Bancach space. Also we have the usual inner product : ...
4
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1answer
46 views

A question about sublinear functionals

Could you please give me hints may leads to prove the following: Let $X$ be a real vector space, $\,p_1,p_2:X\to\mathbb R\,$ be two sublinear functionals, and $\,f:X\to\mathbb R\,$ be a linear ...
2
votes
1answer
30 views

Spectral convergence of coefficients of a Fourier series

I have seen claims that if a smooth function $f(x)$ is represented by its Fourier series, $f(x)=\sum_{n=-\infty}^\infty a_ne^{i(nt)}$, then as $|n|\rightarrow\infty$, then $|a_n|\rightarrow 0$ ...
4
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1answer
28 views

Extension of the limit operator on $l^\infty$

Let $l^\infty = \{x\in \mathbb{R}^\mathbb{N}\colon \sup_{n\in \mathbb{N}}|x_n|<\infty\}$ and the subspace $C \subseteq l^\infty$ given by the convergent sequences. We consider the linear operator ...
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1answer
25 views

Is this a valid operator norm?

The "norm" (yet to be proved or disproved) defined for a matrix $A \in \mathbb{C}^{m\times n}$ by $\|A\|=\max_{i,j}|A_{i,j}|$. Is $\|\cdot\|$ a valid operator norm? (I think it is. As it satisfies ...
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0answers
26 views

Definition of reflexive Banach spaces

I'm trying to understand the definition of reflexive spaces. I wrote in my notes: If $Y$ is reflexive then for all $\eta\in Y^{**}$, $f\in Y^*$, $\exists y\in Y$ where $\eta(f) = f(y)$. My question ...
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0answers
31 views

If $T_n:H\to H$ ($n=1,2,\dots$) are normal linear operators and $T_n\to T$, show that $T$ is a normal linear operator. [on hold]

If $T_n:H\to H$ ($n=1,2,\dots$) are normal linear operators and $T_n\to T$, show that $T$ is a normal linear operator. This seems obvious, so I don't know how to go about showing it. If $T_n:H\to H$ ...
1
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2answers
53 views

About the adjoint operator and weak operator topology.

Let $X,Y$ be Banach spaces. Let $\lbrace{S_n\rbrace}\subset\mathcal{L}(X,Y)$, and $T\in\mathcal{L}(X,Y)$, such that $S_n\xrightarrow[n\to\infty]{WOT}T$, that is: $$\langle ...
1
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2answers
12 views

Show that the matrix $(a_{j,k})_{j,k\in \mathbb{N}}$ induces a bounded operator on $\ell^2$.

I have a matrix $(a_{j,k})_{j,k\in\mathbb{N}}$ given by: $ a_{j,k} = \dfrac{1 -e^{-jk}}{jk + 1}$ and I need to show that this induces a bounded operator on $\ell^2$. I'm pretty sure Schur's test is ...
0
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1answer
13 views

what is permutation invariant sequence space

what is permutation invariant sequence spaces? And why $c_{00}$ is the smallest permutation invariant sequence space?