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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26 views
help with a prove [duplicate]
consider the complex linear spaces $ l_{1}, l_{\infty }$ and the subspace $ c_{0}$ of $l_{\infty }$ sequences consisting of $ \left ( x_{n} \right )_{n\in \mathbb{N}}$ such that $\lim_{x \to 0}=0$.
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1answer
54 views
Question about space of sequences.
Consider the complex linear spaces $l_{1}, l_{\infty }$ and the subspace $c_{0}$ of $l_{\infty }$ sequences consisting of $\left ( x_{n} \right )_{n\in \mathbb{N}}$ such that $\lim_{x \to 0}=0.$
Show ...
2
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0answers
32 views
Is there a bijection there?
Let $X$ be a normed vector space and $T$ a subset of $X^{\prime} = \mathcal{L}(X,\mathbb{R})$. Then define the set:
$$^{\circ}T\ :=\ \{\;x\in X\ :\ F(x)=0,\ \forall\ F\in T\;\}.$$
(When) Is possible ...
9
votes
1answer
66 views
Criteria of compactness of an operator
Suppose $K$ is a linear operator in a separable Hilbert space $H$ such that for any Hilbert basis $\{e_i\}$ of $H$ we have $\lim_{i,j \to \infty} (Ke_i,e_j) = 0$.
Is it true that $K$ is compact?
...
2
votes
1answer
38 views
Convergence of coordinates to zero
Consider a normed finite-dimensional vector space $V$ with some norm $|| .||$
Say a sequence of vectors in this vector space $v_m \rightarrow 0$ where $0$ is the zero vector.
Let ...
1
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0answers
23 views
Why is it true that there are no resonances for Schrodinger operator when dimension is $\geq 5$
For a Schrodinger operator $H=-\Delta+V$, we say that the zero is a resonance of $H$ if the quation $Hu=0$ has a solution $u\notin L^2(\mathbb{R}^n)$ such that ...
3
votes
1answer
31 views
About completeness of $l^{\infty}$ with respect to sup norm
Let $l^{\infty}$ be the space of all bounded sequences of real numbers $(x_n)_{n =1}^{\infty}$ with the sup norm. I have to show that $l^{\infty}$ is complete with respect to this norm.
Proof: In the ...
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votes
1answer
34 views
Inequality for embedding in Sobolev space
For $\Omega=(0,1). $Prove that there exists $M>0$ such that
$$||u||_{C^0(\overline{\Omega})}\le M||u||_{H^1(\Omega)}$$
for all $u\in H^1(\Omega).$
1
vote
0answers
23 views
$\ast$-homomorphism
Let $\phi: C(X,M_{4}(\mathbb{C})) \rightarrow C(Y,M_{8}(\mathbb{C})) $ be a $\ast$-homomorphism where $X$ and $Y$ are compact Hausdorff spaces. Let $M_{2}(\mathbb{C})$ be the C*-subalgebra of ...
3
votes
1answer
42 views
Measurable function of a transformation still measurable
I'm looking for a maybe simpler or more elemental proof of the following statement:
Let $f:\mathbb{R^n}\to \mathbb{R}$ be (Lebesgue-)measurable.
Then $F:\mathbb{R^{2n}}\to \mathbb{R}$ such that ...
2
votes
1answer
25 views
Gateaux and Fréchet differentials in $\ell^1$
I am in trouble trying to solve the following:
Let $X = \ell^1$ with the canonical norm and let $f \colon \ell^1 \ni x\mapsto \Vert x \Vert$. Then $f$ is Gateaux differentiable at a point $x = ...
3
votes
1answer
37 views
Can the orbits of a semigroup touch without one being included in the other?
Question. Let $(S(t))_{t \ge 0}$ be a continuous semigroup of linear operators on some Banach space $X$. Might there exist $f, g\in X$ and $0<t_0<t_1$ such that
...
2
votes
0answers
40 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
3answers
38 views
Compact subsets of function spaces, geometry
The subset is called compact when every open cover contains a finite subcover. In Euclidean spaces, it is easy to visualize this by imagining some open ball that contains this set, thinking about the ...
2
votes
1answer
63 views
Why is there an element with infinite spectrum in a commutative Banach algebra with infinitely many characters?
Let $A$ be a commutative Banach algebra such that set of all characters is infinite. I want to prove that there exists an element in $A$ such that its spectrum is infinite.
2
votes
1answer
35 views
If for every $x_n$ such that $x_n \rightarrow x$, there exists a $x_{n_k}$ such that $Tx_{n_k} = Tx$, is $T$ continuous?
Let $X$ and $Y$ be Banach spaces and $T$ be the (possibly nonlinear) map $T:X \rightarrow Y$. $T$ is continuous if for every $x_n \in X$ such that $x_n \rightarrow x$, then $Tx_n \rightarrow Tx$. Is ...
1
vote
1answer
29 views
Subalgebras of certain C*-algebras
Let $A$ be a C*-subalgebra of $C(X, M_{n}(\mathbb{C}))$ where $X$ is a compact Hausdorff space, does it follow that $A$ is isomorphic to $C(Y, M_{m}(\mathbb{C}))$ for some $Y\subseteqq X$ and ...
4
votes
3answers
89 views
Is there an easy way to see that all derivatives are bounded?
Show that all derivatives of $f:\mathbb{R}\to\mathbb{R}$ given by $$f(x):=\frac{1}{\sqrt{x^2+1}+1}$$ are bounded.
It's easy to see that all derivatives are continuous. So the only potential ...
0
votes
0answers
22 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
23 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
31 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
25 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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1answer
46 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
78 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
41 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
46 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
33 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
35 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})},$$
for all $k\geq 1,1\leq j\leq k$.
This serves as an example of ...
1
vote
1answer
48 views
Convex functions on real vector spaces
So I'm trying to solve the following problem,
Suppose that $f$ is a non-zero convex real-valued function on a real vector space $V$ with $f(0) = 0$
Show that there is a linear functional $g$ on $V$ ...
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votes
0answers
39 views
A $*$-homomorphism from the CAR algebra to $\mathfrak B(\mathcal H)$
Could a $*$-homomorphism $\pi:\text{CAR}\to\mathfrak B(\mathcal H)$ exist (with $\mathcal H$ separable) such that there is a compact and positive element $h\in\mathcal K$ commuting with the image of ...
2
votes
1answer
37 views
Are spaces of finite sequences nuclear?
Let $I$ be some index set and $c_{00}$ the set of functions $c$ from $I$ to $\mathbb{C}$ such that $c(i) \neq 0$ for only finitely many $i \in I$.
Let this space carry the locally conves topology ...
0
votes
1answer
44 views
Spectrum of a unitary
I have a unitary element $v$ in $C(S^{1}, \mathbb{C})$ with full spectrum (the whole circle). Is it possible to construct another unitary $u$ in $C(S^{1}, \mathbb{C})$ out of $v$ such that the ...
-1
votes
0answers
22 views
Proving an operator is self-adjoint
Prove the operator $L$ is self-adjoint.
$$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),$$
where ...
0
votes
1answer
35 views
Show reflexive normed vector space is a Banach space
$X$ is a normed vector space. Assume $X$ is reflexive, then $X$ must be a Banach space.
I guess we only need to show any Cauchy sequence is convergent in $X$.
2
votes
1answer
48 views
Weak $L^{p}$ spaces are quasi-normed?
Let $(X,\mathcal{B}, \mu)$ be a measure space. Then for $0< p < \infty$ by definition
$L^{p,\infty}(X,\mathcal{B}, \mu)$ is the class of all measureable functions $f$ such that
...
1
vote
1answer
27 views
Lipschitz condition normed vector space
Am I right that $g: C^{1}[0,2],||*||_{C^1[0,2]} \rightarrow \mathbb{R}$ with $g(f)=f'(1)$ satisfies a Lipschitz condition?
Cause $|f'(1)-h'(1)|=|g(f)-g(h)|\le \text{max} |f-h|+ \text{max} |f'-h'|$, ...
3
votes
1answer
51 views
Geometric intuition behind the Uniform Boundedness Principle
Is there a way to visualize why the Uniform Boundedness Principle should be true? I understand the statement of the theorem but I'm having a hard time seeing a picture of it in my head.
3
votes
2answers
55 views
If $T:X \to Y$ is a linear homeomorphism, is its adjoint $T^*$ a linear homeomorphism?
$X$ and $Y$ denote Hilbert spaces. If $T:X \to Y$ is a linear homeomorphism, is its adjoint $T^*$ a linear homeomorphism? Homeomorphism means continuous map with continuous inverse.
I think the ...
1
vote
1answer
29 views
Absolute value of an element in a C*-algebra
Is absolute value of a partial isometry a partial isometry itself?
1
vote
1answer
51 views
Why is ess sup $f$ not ess max $f$?
Consider a measure space $(X,\Sigma\,\mu)$. Given that one can easily prove that, $\mu$-a.e., $f \leq \text{ess} \sup_X f$, why is the notation not simply "$ \text{ess} \max_X f$"?
(Here $\text{ess} ...
0
votes
0answers
30 views
Conclusion from vanishing $L^2$ scalar product
Consider the equation
$$\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
18 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
vote
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
31 views
Resolvent of a restriction of a dual operator
UPDATE: After a couple more days of thinking maybe this is however true in general? Is it a known theory?
Theorem(?). Let $A\colon D(A) \to X$ be a linear, densely defined operator in a Banach ...
3
votes
0answers
37 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
21 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
41 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
30 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 $\|x_0\|\leq M$.
