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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0answers
8 views
A property involving a polar of a set
Let $X$ be LCTVS and let $X'$ be its topological dual. Let $A\subseteq X$. Suppose that $x\in X$ and satisfies the following property:
$$|x'(x)|<1 \mbox{ for all } x'\in A^0$$
where $A^0$ is a ...
0
votes
0answers
16 views
Continuity of a certain matrix-like function
Let $X$ be a finite set and let $M$ be a space of all probability measures on $X$. Let $f:X\to\Bbb R^{m\times m}$ be a random matrix and consider a function $c:M\to\Bbb R$ defined as
$$
c(\mu) := ...
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votes
1answer
19 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
1
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1answer
23 views
Find flow of the vector field $\overrightarrow{\operatorname{rot}F}$
We've given 5 points in $\mathbb{R}^3$: $A=(1,0,0)$, $B=(0,1,0)$, $C=(0,0,1)$, $D=(1,1,0)$, $E= (1,1,1)$. We have a surface $S$ given by triangles $ADE, DBE, BCE, CAE$. We have a vector field: ...
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votes
1answer
38 views
convergence in measure does not imply weak convergence
Suppose $\sup_n\|f_n\|_1<\infty$ and $f_n\rightarrow f$ a.e.. However it is not necessary that $f_n\rightarrow f$ weakly in $L^1$.
Can someone raise an example?
Thanks in advance.
3
votes
2answers
58 views
Definition of convergence in $C^\infty(\Omega)$
I am not convinced or may be don't understand, the way they define convergence and then topology as a consequence of convergence.
$\Omega$ is open subset of $\Bbb R^n.$Define standard topology on ...
1
vote
1answer
38 views
About compact operator
When seeing a proof of Fredholm's alternative I don't get the following:
Let $T$ be compact from a Banach space $X$ to itself, and $\lambda \neq 0$. Define $S=I-T$, $S^k$ its $k$-th power and ...
4
votes
3answers
42 views
Introductory/Intuitive Functional Analysis Book
Can you recommend a gentle introduction to the abstract thinking and motivation of functional analysis? I'm looking for a book that holds you by the hand and shows the details of exercises, etc.
...
2
votes
2answers
31 views
Differentiable Operator Continuous?
Consider the space $C^{\infty}[a,b]$ with norm $||f|| = max_{[0,1]} |f(x)|$, with $f\in C^{\infty}[a,b]$. Is the differentiation operator $\frac{d}{dx}$ continuous on $C^{\infty}[a,b]$?
I'm very ...
0
votes
1answer
33 views
Can't establish a lower bound on a supremum
I have a sequence of functions $f_{k,j}:[0,1]\to\mathbb{R}$ defined by
$$f_{k,j} = k^{\frac{1}{p}}\chi_{(\frac{j-1}{k},\frac{j}{k})},$$
for all $k\geq 1,1\leq j\leq k$.
This serves as an example of ...
1
vote
0answers
33 views
Convex functions on real vector spaces
So I'm trying to solve the following problem,
Suppose that $f$ is a non-zero convex real-valued function on a real vector space $V$ with $f(0) = 0$
Show that there is a linear functional $g$ on $V$ ...
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votes
0answers
18 views
A $*$-homomorphism from the CAR algebra to $\mathfrak B(\mathcal H)$
Could a $*$-homomorphism $\pi:\text{CAR}\to\mathfrak B(\mathcal H)$ exist (with $\mathcal H$ separable) such that there is a compact and positive element $h\in\mathcal K$ commuting with the image of ...
2
votes
1answer
35 views
Are spaces of finite sequences nuclear?
Let $I$ be some index set and $c_{00}$ the set of functions $c$ from $I$ to $\mathbb{C}$ such that $c(i) \neq 0$ for only finitely many $i \in I$.
Let this space carry the locally conves topology ...
0
votes
0answers
29 views
Spectrum of a unitary
I have a unitary element $v$ in $C(S^{1}, \mathbb{C})$ with full spectrum (the whole circle). Is it possible to construct another unitary $u$ in $C(S^{1}, \mathbb{C})$ out of $v$ such that the ...
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votes
0answers
21 views
Proving an operator is self-adjoint
Prove the operator $L$ is self-adjoint.
$$L: y''-q(x)y(x)=-\lambda y(x) , x\in[0,\pi],$$
$$\lambda(y'0)-hy(0))=h_{1}y'(0)-h_{2}y(0),$$
$$\lambda(y'(\pi)+H y(\pi))=H_{1}y'(\pi)+H_{2}y(\pi),$$
where ...
0
votes
1answer
34 views
Show reflexive normed vector space is a Banach space
$X$ is a normed vector space. Assume $X$ is reflexive, then $X$ must be a Banach space.
I guess we only need to show any Cauchy sequence is convergent in $X$.
2
votes
1answer
48 views
Weak $L^{p}$ spaces are quasi-normed?
Let $(X,\mathcal{B}, \mu)$ be a measure space. Then for $0< p < \infty$ by definition
$L^{p,\infty}(X,\mathcal{B}, \mu)$ is the class of all measureable functions $f$ such that
...
1
vote
1answer
26 views
Lipschitz condition normed vector space
Am I right that $g: C^{1}[0,2],||*||_{C^1[0,2]} \rightarrow \mathbb{R}$ with $g(f)=f'(1)$ satisfies a Lipschitz condition?
Cause $|f'(1)-h'(1)|=|g(f)-g(h)|\le \text{max} |f-h|+ \text{max} |f'-h'|$, ...
3
votes
1answer
47 views
Geometric intuition behind the Uniform Boundedness Principle
Is there a way to visualize why the Uniform Boundedness Principle should be true? I understand the statement of the theorem but I'm having a hard time seeing a picture of it in my head.
2
votes
2answers
52 views
If $T:X \to Y$ is a linear homeomorphism, is its adjoint $T^*$ a linear homeomorphism?
$X$ and $Y$ denote Hilbert spaces. If $T:X \to Y$ is a linear homeomorphism, is its adjoint $T^*$ a linear homeomorphism? Homeomorphism means continuous map with continuous inverse.
I think the ...
1
vote
1answer
26 views
Absolute value of an element in a C*-algebra
Is absolute value of a partial isometry a partial isometry itself?
1
vote
1answer
48 views
Why is ess sup $f$ not ess max $f$?
Consider a measure space $(X,\Sigma\,\mu)$. Given that one can easily prove that, $\mu$-a.e., $f \leq \text{ess} \sup_X f$, why is the notation not simply "$ \text{ess} \max_X f$"?
(Here $\text{ess} ...
0
votes
0answers
28 views
Conclusion from vanishing $L^2$ scalar product
Consider the equation
$$\int_{\mathbb R}\overline{\phi(x)}e^{iax}\psi(x+b) dx = 0, \qquad a,b \in \mathbb R$$
i.e. orthogonality of $L^2$ functions.
How can one conclude that $\phi(x)$ and ...
1
vote
1answer
39 views
Why this space is not a complete space with this norm
Show that the space $C_0(\mathbb{R})$ of all the real continuous functions $f:\mathbb{R} \to \mathbb{R}$ with compact support is not a complete space with the norm $||f||= \sup_{t∈ \mathbb{R}}|f(t)|$.
...
1
vote
1answer
16 views
Finite-dimensional irreducible representations
How do we show that a finite-dimensional $*$-representation of a $C^{*}$-algebra is unitary equivalent to a direct sum of irreducible representations?
1
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1answer
39 views
Is a Banach Space
Show that the vector space, $P_n$, of all the real polynomial functions of degree less than n, is a Banach Space for any norm define.
I think if I prove that $P_{n}$ is a Banach Space with the norm ...
0
votes
0answers
14 views
Resolvent of a restriction of a dual operator
Let me begin with some well known theory. Let $ A\colon X \supset D(A) \to X $ be a linear, densely defined (that is $\overline{D(A)} = X$) operator in a Banach space $ X $. Assume that for all $ ...
3
votes
0answers
33 views
Fredholm operators and Compact operators
Suppose $X$ be an infinite dimensional Banach space. How to prove that:
$A$ and $B$ are two Fredholm operator on $X$, if
$\mathrm{index}(A)=\mathrm{index}(B)$, then there exists an invertible ...
0
votes
0answers
18 views
The invertible Toeplitz operators on H^2 space
Suppose $ϕ$ be a real-valued function. I want to prove that the Toeplitz operator $T_ϕ$ is invertible if the constant function $1$ is in the range of $T_ϕ$.
There is a function $f\in H^2$ such that ...
0
votes
1answer
39 views
Multiplication in $\mathcal D'(R)$.
I was reading in my text book that multiplication of elements of $\mathcal D'(R)$ is not a continuous function although it is defined and can be seen from the convergence of $\sin(nx)$ to $0$ as $n$ ...
0
votes
1answer
28 views
Quadratic Functions
Consider the strictly convex quadratic function $f(x) = \frac{1}{2}x^tPx - q^tx + r,$ where $P \in \mathbb{R}^{n \times n}$ is a positive definite matrix, $q \in \mathbb{R}^n$ and $r \in \mathbb{R}.$ ...
-1
votes
0answers
29 views
weak convergence in reflexive normed space
$X$ is a reflexive normed space. $\{x_n\}\subset X$ is a sequence bounded by $M$. Show that there exist a subsequence $\{x_{n_k}\}$ such that $x_n$ converges to $x_0$ weakly and $\|x_0\|\leq M$.
3
votes
3answers
34 views
Incomplete space
I must show that $C_0(\mathbb{R})$ space of all continuous real functions $f:\mathbb{R} \longrightarrow \mathbb{R}$ with compact support is not a complete space endowed by norm
$\|f\|=\sup\limits_{t ...
1
vote
1answer
51 views
How to find the unknown values in this Numerical Integration type?
Given the following type of numerical integration:
$$I(f)=\int_0^1 f(x) \, dx \approx \frac 12 f(x_{0}) +c_1 f(x_1) $$
a) Find the values of: the coefficient $c_1$ and points $x_0$ and $x_1$ so ...
0
votes
1answer
62 views
Does convergence in $C(\Bbb R)$ imply convergence in $\mathcal D'(\Bbb R)$
Sequence of continuous functions $(f_k)$ which converge to $f$ in $C(\Bbb R)$. Does it imply that $f_k$ will converge to $f$ in $\mathcal D'(\Bbb R)$ as well? Or we need something more to make this ...
2
votes
1answer
44 views
Norm of an operator
How to find the norm of the following operator
$$
A:\ell_p\to\ell_p:(x_n)\mapsto\left(n^{-1}\sum\limits_{k=1}^n k^{-1/2} x_k\right)
$$
Any help is welcome.
2
votes
1answer
28 views
there is no bounded linear functional on $ H$
let $ H= L^2[0,1]$ and $ C^1 $ be the set of all continuouse functions on $ [0,1] $ that have continuouse derivative.Let $ t \in [0,1] $ and define $ L: C^1 \longrightarrow F $ by $ L (h)= h'(t) $. ...
1
vote
0answers
22 views
Hahn Banach to get linear functional bounded by sub/superlinear functionals
I am working in a real vector space $V$. I have seen it written that if I have a sublinear functional $p$ and a superlinear functional $q$ such that $q \le p$ then there exists some linear functional ...
-1
votes
0answers
17 views
Help with Toeplitz operators applications. [closed]
I am trying to find a physics problem which solution involves Toeplitz operators.
0
votes
0answers
27 views
Commuting distributions with integration
Let $f : \mathbb R^n \times \mathbb R^m \rightarrow\mathbb R$ be a smooth function, $L^1$ function of two variables such that $x\mapsto f(x,y)$ is Schwarz for each fixed $y$. Let $u$ be a compactly ...
1
vote
1answer
37 views
Diffeomorphic surfaces and Jacobian
Suppose $S$ and $T$ are bounded (open) surfaces in $\mathbb{R}^n.$ Let them have boundary $\partial S$ and $\partial T$.
Suppose $F:S \to T$ is a $C^k$ diffeomorphism.
Under what conditions on $F$ ...
1
vote
2answers
63 views
If $K$ is compact, then $C(K,\mathbb{R}^n)$ is a Banach space under the norm $\|f\|=\sup_{x\in K} \|f(x)\|$
Let $K$ be a topological space that is compact. Show that the space $C(K,\mathbb{R}^n)$ of all the continuous functions $f:K\to\mathbb{R}^n$ is a Banach space with the norm $\|f\|=\sup_{x\in K} ...
0
votes
0answers
16 views
Definition of $C^1_c((a,b);H)$
Can I define $C^1_c((a,b);H)$ as space of compactly-supported continuous functions from $[a,b]$ to a Banach space $H$ with compactly supported derivative (that may not be continuous)? So basically I ...
2
votes
1answer
47 views
Function spaces and transitive group actions
Note: this question is really a subquestion of this one, but I decided to ask it separately since it seems it might be attacked first.
Let $B$ be a topological space and $G$ a topological group ...
1
vote
1answer
20 views
Is this function measurable? Something to do with Bochner space and norms.
Suppose $f:[0,T]\to X$ is a measurable map where $X$ is Hilbert space. Suppose also that $R(t):X \to X^*$ is an isometric isomorphism with
$$\lVert R(t)f(t)\rVert_{X^*} = \lVert f(t) \rVert_X$$
also ...
0
votes
1answer
59 views
Show $\|f\|_p\leq \lim\inf\|f_n\|$
$\Omega$ is a bounded domain of $\mathbb R^n$. If $\{f_n\}\subset L^p(\Omega)$ and $f_n\rightarrow f\in L^p(\Omega)$ weakly, then
$$\|f\|_p\leq \lim_{n\rightarrow\infty}\inf\|f_n\|$$
2
votes
1answer
66 views
Bounded sequence in Hilbert space contains weak convergent subsequence
In Hilbert space $H$, $\{x_n\}$ is a bounded sequence then it has a weak convergent subsequence.
Is there any short proof? Thanks a lot.
2
votes
1answer
21 views
Selfadjoint operator $\Rightarrow$ Idempotent Operator?
If $P\in\mathcal{L}(H,H)$, with $H$ a Hilbert space, such that $P = P^*$, Is possible to show that $P^2 = P$?
If that is possible, then $P$ is a projection operator, right?
Thanks in advance.
2
votes
0answers
45 views
Norm inequalities in a reflexive space
I am reading an article about reflexive spaces, with a specific example. The article mentions inequalities that I haven't been able to get around to. Here's the setup.
The space $X = (\prod_n ...
2
votes
1answer
29 views
an example to show separability of a Banach space does not imply separability of the dual space
$X$ is a Banach space and it is separable, is there any simple counterexample to show the dual space $X^\ast$ is not separable?
