6
votes
1answer
103 views

Non-examples for the Kato-Rellich Theorem

The Kato-Rellich Theorem is a classical result stating that if $A,B$ are unbounded operators on a Hilbert space with $A$ self-adjoint, $B$ symmetric, $\mathcal D (A)\subset \mathcal D(B)$ and $$ ...
1
vote
1answer
30 views

Counterexample for Palais-Smale condition

I have trouble proving that functional $I:H\to\mathbb{R}$ given by $$I(u)=\frac{1}{2}\|u\|^2-\frac{1}{2}(u,f)^2$$ does not satisfy Palais-Smale condition if $\|f\|=1$. I managed to prove that when ...
0
votes
1answer
24 views

Various convergences in the space of bounded operators

Could you please help me to find some classical (counter)examples in functional analysis? Let $X$ and $Y$ be some normed spaces over $\mathbb{C}$. By $\mathcal{B}(X,Y)$ we denote the space of bounded ...
1
vote
1answer
48 views

What is wrong with this “counterexample” of boundedness of weakly convergent sequences?

Weakly convergence sequences $\{u_n\}$ in a Hilbert space $H$ are bounded. Here is an attempted "counterexample". What is wrong with this? Let $H = \ell_2(\mathbb{N})$, and let $\{e_n\}$ be the ...
2
votes
2answers
53 views

Topological vector spaces vs. locally convex spaces

I'm taking a course on locally convex spaces and our lecturer mentioned that these form the most general collection of spaces on which one can still prove interesting theorems (like Hahn-Banach - ...
2
votes
1answer
47 views

Question about compact operator

So here is my question, Let $H$ be a Hilbert-space and $K:H\rightarrow H$ a compact operator. I know that if $K$ is self adjoint, then it has one eigenvalue $\mu$ such that $|\mu|=||K||$. Can some ...
1
vote
1answer
50 views

Counterexample to Marcinkiewicz

I have a version of Marcinkiewicz: Let $(X,\mu)$ and $(Y,\nu)$ be measure spaces and let $1<p_1 \leq \infty$. Suppose that $T$ is a mapping from $L^1(X,\mu) + L^{p_1}(X,\mu)$ to $\mu$- measurable ...
4
votes
2answers
61 views

Question about a counterexample concerning compact operators

Does anybody know if the following is true, Let $H$ be an infinite dimensional Hilbert-space and $K:H\rightarrow H$ a compact operator. Then if $|\mathrm{spec}(K)|<\infty$ i.e the spectrum is ...
1
vote
1answer
39 views

Counterexamples for L1-convergence not implying L2-convergence and vice versa

I am trying to find counterexamples for the following statements. Let $\{f_n\}$ be a sequence in $L^1(\mathbb{R}^d) \cap L^2(\mathbb{R}^d)$, and let $f$ also be in $L^1(\mathbb{R}^d) \cap ...
1
vote
1answer
40 views

Compact surjective non injective operator

Let $X$ be an infinite dimensional Banach space. I know that every compact operator $A$ is not bijective or $0\in\sigma(A)$. Fox example the compact operator $A$ defined on $X=C([0,1],\mathbb{R})$ ...
1
vote
1answer
56 views

weak star convergence of signed measures vs convergence in Fortet-Mourier norm

There is a norm for signed measures given by $$||\mu||_{FM}=\sup_{f\in \mathrm{Lip}_1(X),|f(x)|\leq 1}\langle f,\mu\rangle.$$ This is usually called Fortet-Mourier norm (or more often metric, but it ...
4
votes
1answer
86 views

Caught in the net

I'm reading through some notes one locally convex spaces ("lcs" from now on) analysis and there the following version of the Banach-Steinhaus theorem is given Theorem (Banach-Steinhaus) $\quad$ ...
1
vote
0answers
69 views

Maximal monotone operator without convex domain?

I'm looking for an example of a (multi-valued) maximal monotone operator $A$ mapping a Banach space $X$ into its dual $X^*$ such that the domain $D(A)=\{x\in X: Ax\neq\emptyset\}$ is not convex. ...
3
votes
1answer
57 views

Function in $H^1$, but not continuous

By the Sobolev embedding theorem, if $\Omega$ is bounded, $H^s(\Omega)\subset C(\Omega)$ for $s>1$, in $\mathbb R^2$. Where I can find a counterexample (if one exists) for the case $s=1$? I mean a ...
0
votes
1answer
47 views

Counterexample of Separation Theorem in topological vector space

The Separation Theorem states that: If $A$ and $B$ be two disjoint convex subsets in a vector space $X$ and one of them has nonempty core (algebraic interior) then there exists a linear functional ...
0
votes
1answer
34 views

Closed, bounded and convex subset in $X^*$ but not $w^*$ closed.

Banach-Alaoglu states that: If $X$ is a topological vector space then the polar of any neighborhood of the origin is $\sigma(X^*,X)$ compact. Especially, if $X$ is a norm linear space then the closed ...
1
vote
1answer
41 views

Condition to separability of a Banach space.

I am trying to prove the following statement: Let X be a Banach space and $X^{*}$ its topological dual space. If there exists a countable family of functions $(f_{n})_{n} \subset X^{*}$ such that ...
1
vote
1answer
50 views

Counterexample of separation theorem

I'm trying to know a counterexample for separation theorem: If $A$ and $B$ are two disjoint convex set in a topological vector space $X$, one of them has nonempty interior, then there exists $f\in ...
1
vote
0answers
36 views

nearest point and closed complement of a subspace in norm spaces

It is well-known that in any Hilbert space $H$, each closed subspace $Y$ admits a closed complement $Y^\perp$. This result also implies that there exists a best approximation point to $Y$ for any ...
0
votes
1answer
38 views

Projection onto a subset in norm space

Let $X$ be a norm space, $S$ be a subset of $X$, for each $x\in X$, denote the set of projection from $x$ to $S$ by $\Pi(x; S):=\{s\in S: \ \|x-s\| =d(x;S)\}$, where $d(x;S):=\inf\{\|x-s\|: \ s\in ...
3
votes
0answers
136 views

Why this is not a Banach space

When reading about functional analysis I encountered the following example of a Banach space: $ C^1 ([0,1])$ endowed with the norm $\|f\| = \|f\|_\infty + \|f'\|_\infty$. where $\|\cdot\|_\infty$ ...
0
votes
1answer
62 views

About a counterexample for the Open Mapping Theorem

I want to ask some thing about a counterexample for the open mapping theorem: Find a discontinuous linear mapping $T: \ X \to Y$ such that $T(X)=Y$ and $X,Y$ are Banach but $T$ is not open. I find ...
3
votes
3answers
108 views

Necessity of completeness of the inner product space in Riesz representation theorem

I wanted to find a counter example to show that the completeness of the inner product space is necessary in Riesz representation theorem. Please give an example of a bounded linear functional $T$ on ...
1
vote
1answer
72 views

Is every Hilbert space an $L^2$ space

Let $H$ be any Hilbert space. Must there exist a measure space $(X,\scr{M},\mu)$ such that we have a Hilbert space isomorphism: $$H\cong L^2(\mu)$$ Thank you
5
votes
1answer
50 views

$L^1(\mathbb{R}^n)$ functions not in $\mathcal{H}^1(\mathbb{R}^n)$

I am wondering how to imagine the Hardy space on $\mathcal{H}^1(\mathbb{R}^n)$ and in particular what sort of functions are in $L^1(\mathbb{R}^n)\backslash\mathcal{H}^1(\mathbb{R}^n)$. Furthermore, is ...
4
votes
1answer
335 views

Counterexample for the Open Mapping Theorem

I would like to ask a counterexample for the open mapping theorem: Find a discontinuous linear mapping $T:X \to Y$ such that $T(X)=Y$ and $X,\;Y$ are Banach but $T$ is not open. Could you help me ...
5
votes
2answers
305 views

Operator with invertible adjoint

Let $X, Y$ be normed spaces and $T : X \to Y$ bounded and linear, such that its adjoint $T^* : X^* \to Y^*$ is boundedly invertible. If $X$ and $Y$ are Banach spaces, then $T$ is also boundedly ...
2
votes
0answers
94 views

Is there a “natural” norm in the space of all sequences?

Let $X$ be $\mathbb{R}^\mathbb{N}$, the space of all real-valued sequences. Is there a natural norm in this space?
1
vote
1answer
145 views

Non-commuting projection operators on a Hilbert space

Let $H$ be a separable Hilbert space. Can you provide an example of 3 orthogonal projection operators which are mutually non-commuting?
1
vote
1answer
175 views

A counter example of best approximation

Construct a point $f\in C[0,1]$ and a closed subspace $V\subset C[0,1]$ such that $f$ does not have a best approximation in $V$. Definition: $C[0,1]$ is the set of countinous function with the norm ...
3
votes
3answers
111 views

How common is it for a densely-defined linear functional to be closed?

I've always held the vague belief that any densely-defined operator encountered "in nature", if it isn't bounded, is probably at least closable. But, today I noticed the following thing: Consider ...
5
votes
1answer
111 views

Example of a singular element which is not a topological divisor of zero

We know that every topological divisor of zero in a commutative Banach algebra is singular. I need an example of a singular element which is not a topological divisor of zero.
7
votes
3answers
526 views

Banach Algebra counterexample

Can someone give me an exemple of a Banach Algebra $\mathbb{A}$, for which there is no isometric representation in a Hilbert Space ? (with proof or references to proof) Thank you very much :)
1
vote
3answers
1k views

Examples of metric spaces which are not normed linear spaces?

Give an example of a metric space which is not a normed linear space. Justify your example.
2
votes
1answer
196 views

A semisimple commutative Banach algebra with a non-semisimple quotient

I am looking for an example of a semisimple commutative Banach algebra which admits a non-semisimple quotient. Attempt from the comments: "I take $A$ to be the algebra of all continuously ...
1
vote
1answer
42 views

Real commutative Banach algebra with identity

I am looking for example of real commutative Banach algebra with identity which does not admit a nonzero real multiplicative linear functional
2
votes
0answers
54 views

Inequality change in $\mathbb{E}[ \max |\cdot|] $ due to $\max$

Let $m$ be a probability measure on $W \subseteq \mathbb{R}^p$, so that $m(W)=1$. Find $m$, a locally-bounded function $f:\mathbb{R}^n \times \mathbb{R}^m \times \mathbb{R}^p \rightarrow ...
13
votes
1answer
654 views

Open Mapping Theorem: counterexample

The Open Mapping Theorem says that a linear continuous surjection between Banach spaces is an open mapping. I am writing some lecture notes on the Open Mapping Theorem. I guess it would be nice to ...
3
votes
1answer
110 views

How to construct an “explicit” element of $(\ell^\infty(\mathbb N))^* \setminus \ell^1(\mathbb N)$? [duplicate]

Possible Duplicate: Nonnegative linear functionals over $l^\infty$ An explicit functional in $(l^\infty)^*$ not induced by an element of $l^1$? Everything is in the title: How to ...
3
votes
0answers
79 views

Application of a result on some bounded functionals on a subspace of $C([0,1])$

The following result was proved in a previous post: Bounded functionals on Banach spaces. Let $(X, \|.\|)$ be a Banach space such that $X \subset C([0,1]) $ For every $r\in \mathbb{Q}\cap[0,1], ...
1
vote
2answers
250 views

Weierstrass M-test proof?

Let (X,d) be a metric space. For each n $\epsilon$ N let $g_n$:X$\rightarrow$R be a continuous function. Let ($a_n$) be a sequence of positive real numbers such that the series $\sum_n_=_1^\infty a_n$ ...
2
votes
1answer
56 views

So $k^2-\Delta: H_{s+2}\to H_{s}$ is a homeomorphism, but what does that tell us?

For each $t\in\mathbb{R}$, we define the Sobolev space \begin{equation} H_t=\{u\in\mathcal{S}':\int(1+|y|^2)^t|\hat{u}(y)|^2dy<+\infty\}, \end{equation} where $\mathcal{S}'$ is the space of ...
1
vote
1answer
106 views

How to show that linear span in $C[0,1]$ need not be closed [duplicate]

Possible Duplicate: Non-closed subspace of a Banach space Let $X$ be an infinite dimensional normed space over $\mathbb{R}$. I want to find a set of vectors $(x_k)$ such that the linear ...
6
votes
2answers
74 views

Alternate definition for boundedness in a TVS

Let $X$ be a topological vector space over $\mathbb R$ or $\mathbb C$. A subset $B\subset X$ is defined to be bounded if for any open neighborhood $N$ of $0$ there is a number $\lambda>0$ ...
0
votes
2answers
142 views

How to cook up test functions?

Let $\Omega\subset \mathbb{R}$ be open. A test function is a $C^\infty$ function with compact support. This is a rather strong restriction, for instance, no analytic function is a test function. But ...
6
votes
1answer
538 views

Some examples in C* algebras and Banach * algebras

I would like an example of the following things. A Banach * algebra that is not a C* algebra for which there exists a positive linear functional (it takes $x^*x$ to numbers $ \geq 0$) that is not ...
1
vote
1answer
59 views

Nice example where $D^{\alpha}\Lambda_{f}\neq\Lambda_{D^{\alpha}f}$?

Let $\Omega\subset\mathbb{R}^n$ be open and $f$ be a locally integrable function. The distribution associated with $f$, $\Lambda_{f}\in D'(\Omega)$, is defined via \begin{equation} ...
7
votes
2answers
2k views

convolution of a function with itself equals itself

In a homework question, I was asked to show: (1) in $L^1(R)$, if $f*f = f$, then $f$ must be a zero function. (2) In $L^2(R)$, find a function $f*f=f$. I don't know how to proceed. for (1), $f*f=f$ ...
5
votes
2answers
308 views

Square root of compactly supported C-infinity function

Given $u \in \mathcal{C}^\infty_0(\mathbb{R}^n)$, $u \geq 0$ everywhere, is $v(x) = \sqrt{u(x)}$ also in $\mathcal{C}^\infty_0$? It is clear that the only problematic points are the boundary of the ...
6
votes
1answer
338 views

Rainwater theorem, convergence of nets, initial topology

I've stumbled upon a result called Rainwater's theorem a few times, it seems to be a very useful result in connection with weak convergence in Banach spaces. Rainwater's theorem. Let $X$ be a ...