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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Some question about linear operator on normed space

$1)$ let $X$ and $Y$ be normed space , show that a linear operator $T:X\rightarrow Y$ is bounded if and only if $T$ maps bounded sets in $X$ into bounded sets in $Y$ $2)$show that the operator ...
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1answer
14 views

is $\langle\lim_{n\to \infty}u_n,g\rangle = \lim_{n\to\infty} \langle u,g\rangle $ valid for bounded linear operators?

Suppose M is any linear manifold in H. H is a hilbert space. Define the orthogonal complement of M to be $$M' =\{f \in H | \langle f,g\rangle= 0 ,\forall g\in M\}.$$ To see that M' is a closed ...
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3 views

Singular Spectrum: Techniques?

Given a Hilbert space $\mathcal{H}$. Let the Lebesgue measure be $\lambda$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Denote its spectral measure by: ...
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8 views

Are all linear basis functions a reproducing kernel hilbert space?

Do any linear basis function like for instance linear b-splines form a reproducing kernel hilbert space? is it sufficient for the kernel to be semi-positive definite and have a positive Fourier ...
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7 views

Are not all neighborhoods of $0$ in a locally convex space absorbent?

A locally convex space (LCS) can be defined as a topological vector space (i.e. scalar product and sum are continuous) whose topology is generated by translation of a family of balanced and absorbent ...
2
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1answer
24 views

Point spectrum of operator on $\ell^2$?

Considere the bounded linear operator $S:\ell^2\longrightarrow \ell^2$ given by $$ S(\xi_j)_j:=\left(\frac{\xi_2}{1}, \frac{\xi_3}{2}, \frac{\xi_4}{3}, \ldots\right).$$ How to show the point spectrum ...
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1answer
34 views

Is it or not a continuous embedding?

Please I have this two spaces $$C_{\theta}=\{u\in C(\overline{\Omega}), \sup (|x|^{\theta} |u(x)|)<\infty\}$$ with the norm $\displaystyle||u||_{\theta}=\sup_{\Omega}(|x|^{\theta} |u(x)|)$ and ...
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1answer
19 views

Proof $x\to ||f(x)||_B \in \mathbb{L}^1(X,S,\mu,\mathbb{R})$

Why is the function $g:x\to ||f(x)||_B$ in $\mathbb{L}^1(X,S,\mu,\mathbb{R})$, where $f\in\mathbb{L}^1(X,S,\mu,\mathbb{R})$, and $||\cdot||_B$ is the norm in the Banach space that $f$ maps into? I ...
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1answer
27 views

Closed point problems in $l$-spaces, namely $ l^1$ and $l^\inf$

a) Give an example of a closed convex set $C$ and point $x$ in $l^\infty$ such that the closed point in $C$ to $x$ is not unique. Solution: So I was thinking that a closed convex set would be the ...
3
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2answers
52 views

Cardinality of a basis of an infinite-dimensional vector space

How would you find the cardinality of the basis of $\mathbb{R}$ over $\mathbb{Q}$? Is it countable or uncountable? In general, how do you find the cardinality of a basis of an infinite-dimensional ...
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1answer
28 views

If I want to prove that $M^{\perp}$is a closed

If I want to prove that $M^{\perp}$is a closed Can I say because it is the inverse image of $0$ by continuos function ( projection operator )
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1answer
49 views

How can I prove the following theorem with explanation? please

How can I prove the following theorem with explanation. please For any nonempty subset $M$ of a Hilbert space $H$, the span of $M$ is dense in $H$ if and only if $M^{\perp}=\{0\}$ I read the prove ...
2
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1answer
15 views

Isomorphism of $4$ dimensional hilbert space with tensor product of two dimensional Hilbert space

I want to know what will be the isomorphic map between $\mathbb{C}^4$ over $\mathbb{C}$ to $\mathbb{C}^2\otimes \mathbb{C}^2$ over $\mathbb{C}$, as a $4$ dimensional Hilbert space they are isomorphic ...
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1answer
17 views

ONB of Hilbert dual $H'$

Let $H$ an arbitrary Hilbert space, $\{ e_i \}_{i \in I}$ ONB of $H$. Is there an ONB $\{ e^j \}_{j \in I}$ of the Hilbert dual $H'$, s.t. $e^j(e_i)=\delta_{ij}$? If so, is $\{e_i \otimes e^j\}_{i,j ...
2
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1answer
50 views

Dominated positive operator

I want that if $H$ Hilbert space where $A$, $B$ are positive operators on $H$ Hilbert space, $0 \leq (Ax|x) \leq (Bx | x)$ $\forall x$, does this mean $(A^2x|x) \leq (B^2x|x)$? Thank you
2
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2answers
41 views

Is the sequence of functions, $f_n(x) = x^n$ Cauchy in $C([0,1])$? Is it Cauchy in $L^2([0,1])$?

Is the sequence of functions, $f_n(x) = x^n$ Cauchy in $C([0,1])$? Is it Cauchy in $L^2([0,1])$? I know that it is not Cauchy in the space of continuous functions. I know that the L-space is complete ...
4
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3answers
57 views

A uniform bound by an integrable function for a Fourier series' partial sums.

Consider \begin{equation} \sum\limits_{n=1}^\infty\frac{\cos(nx)}{n}=-\log|2\sin x/2|~~~ \big(x\in(0,2\pi)\big), \end{equation} and its $2\pi$-periodic extension $f$ (for a proof of the above ...
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1answer
27 views

Projection of a non-surjective operator

Apologies for the poor title. What I'm wondering is: Say that we have a non-surjective operator $A:X\rightarrow X$ where $X$ is a Banach space, and the operator is defined in terms of the basis ...
2
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1answer
16 views

Is convolution a coercive bilinear form in $L^2$ -space?

This is one of the problems in functional analysis course I'm having. Suppose $f,g \in L^2(0,10)$. Then define a bilinear form $$ B:(f,g)\mapsto \int_0^{10} f(x)g(10-x) dx. $$ Now I have to find out ...
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0answers
24 views

uniform convergance

Let $\chi={x_{1},x_{2},...} $ a dense subset of R. I need to construct a continuous curve in $X$ whose end points are $x_{1}$ and $x_{m}$ and its domain is [0,1]. We start with $\gamma_{1}:\left[ ...
2
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1answer
29 views

Proving that this space is not Hilbert.

Consider $E$ the space of all the functions defined on $\Bbb R$ which admit a representation of the form $x(t) = \sum_{r \in \Bbb R}^* c_r e^{irt}$, where $\sum^*$ indicates that only a finite number ...
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0answers
22 views

Lummer-Phillips theorem for generator of strongly continuous semigroup

Let $L^{2}([a,b];\mathbb{K}^{n})$ be the standard Lebesgue space with its standard inner product. Consider the operator $A$ defined by $\displaystyle ...
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18 views

Operator norm on the space on linear functions between Euclidean spaces.

*I'm reading a text which has a preliminary section on Linear maps. I have come across a conclusion that I can't seem prove by myself. * Let $Lin(\mathbb{R}^m,\mathbb{R}^n)$ be the space of linear ...
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1answer
16 views

Weak convergence on $C(K)$ space implies pointwise convergence

I'm wondering a following problem: If $f_n$ weakly converges to $f$ on $C(K)$ space, then can we conclude that $f_n(x)$ converges to $f(x)$ for every $x$ in $K$? Why?
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0answers
14 views

convergence in different normed spaces

In my class lecture notes, there is such a lemma. Let $X$ be a vector space over $\mathbb F$ and $\lVert \cdot \rVert_1$, $\lVert \cdot \rVert_2$ be two norms on it. If there is $M > 0$ such that ...
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0answers
15 views

Spectrum of adjoint operators

If $A \subset \mathbb C$, we set $A^* = \{\bar z: z \in A\}$. I want to prove the following theorems. $\rho(T)^* = \rho(T^*)$ and $\sigma(T)^* = \sigma(T^*)$. $\sigma_c(T)^* = \sigma_c(T^*)$. ...
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1answer
21 views

check if a linear operator is bounded

show that $Tf = f(0)$ is not a bounded linear functional on the space of continuous functions measured with the L2 norm, but it is a bounded linear functional if measured using the uniform norm. ...
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1answer
11 views

Diagonal non-compact operator

Suppose we have an operator $I:l_2 \rightarrow l_2$ which is diagonal but not compact. Does that follow: there exists a constant $C$ such that infinite number of diagonal terms $>C$?
3
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54 views

Is this derivative well defined?

Let $\Omega$ be an open bounded domain in $\mathbb{R}^n$. Let $z_1,z_2,\dots,z_k\in \bar \Omega $ be $k$ distinct points. Let $\bf Z$ denote the k-tuple $Z = (z_1,z_2, \dots, z_k)$ Define $R_i = ...
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0answers
11 views

the weak* topology on the closed unit ball in $X^*$ is second countable when $X$ is separable

I am trying to approach this problem by using Urysohn's metrization theorem, that is every Hausdorff second-countable regular space is metrizable. The key obstacle for me is that I don't know how to ...
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1answer
32 views

Can someone please explain this proof of the non-separability of $\ell^ \infty$.

Let $A=\{x_n : x_n \in \ell^\infty\}$ be any arbitrary countable subset $\ell^\infty$. For each $n \in \mathbb{N}$, let $x_n = (a_{n_i})_{i=1}^{\infty}.$ Define $y=(b_i) \in \ell^\infty$ as $$ b_i = ...
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0answers
18 views

weak* topology is not defined by any translation invariant metric when $X$ is infinite dimensional

There is an exercise in Folland's real analysis, page 170 If $X$ is an infinite dimensional Banach space, then the weak* topology is not defined by any translation-invariant metric. He gives a ...
3
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2answers
63 views

Two definitions of spectrums

In Kreyszig's Introductory Functional Analysis Page 371, the point spectrum is defined as $\sigma_p(T)$ such that $R_\lambda(T) = (T - \lambda I)^{-1}$ does not exist. While in my functional ...
3
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1answer
22 views

Frechet metric: troubles understanding $d(x^{(j)},0)\to0\iff x_i^{(j)}\to0$ $\forall i\in\mathbb N$

Consider the metric $$ d(x,y)=\sum_{k=1}^\infty\frac1{2^k}\frac{|x_k-y_k|}{1+|x_k-y_k|} $$on $\mathbb R^{\mathbb N}$ with $x=(x_k)$ and $y=(y_k)$. Let $x^{(j)}\in\mathbb R^{\mathbb N}$ for all ...
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0answers
16 views

Continuity of function mapping $\mathcal{P}(\mathcal{P}(X))$ to $\mathcal{P}(X)$

Given a topological space $Y$, let $\mathcal{P}(Y)$ be the set of all probability measures on $Y$, endowed with the weak* topology. Let $X$ be a topological space (for convenience, it might be Polish ...
3
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1answer
20 views

If $u \in H^{\frac 12}(\Omega)$ and $c \in \mathbb{R}$ is $(u-c)^+ \in H^{\frac 12}(\Omega)$ too?

If $u \in H^{\frac 12}(\Omega)$ and $c$ is a constant, is the function $$(u-c)^+ \in H^{\frac 12}(\Omega)?$$ Here $(x)^+$ is $x$ when $x > 0$ and $0$ otherwise. If it were $H^1$ then it is a true ...
2
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3answers
42 views

Checking if a vector field is conservative

I have three different vector fields and I want to check if they are conservative: $$1)\space \space\vec{f}(\vec{x}):=\frac{1}{||\vec{x}||}\vec{a}, \space \space D=\mathbb R^2 / (\vec{0}), \space ...
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1answer
26 views

One question about measure theory [on hold]

Let $(X,S,μ)$ be a finite measure space $μ(X) < \infty $ and $α$ is a finite positive measure on $S$ If $α(A)=\int_{A}{}hdμ$ where $ h \in L^1(μ)$ Prove that $α<<μ$
4
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2answers
46 views

Checking that $(C[0,1], \|\cdot\|_1)$ is not Banach.

I want to check that $(C[0, 1], \|\cdot\|_1)$ is not a Banach space, where $$\|f\|_1 = \int_0^1 |f(x)|\,{\rm d}x.$$ I took $(f_n)_{n \geq 1}$ a sequence in $C[0, 1]$ given by: $$f_n(x) = ...
2
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0answers
63 views

Examples of $C^*$-algebras in Noncommutative Geometry from A. Connes

Question I am working on $C^*$-algebras and I've been given Alain Connes's book Noncommutative Geometry. I am having troubles with understanding the examples on pages 91-93 (86-88 in the printed ...
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1answer
21 views

Sobolev/Lebesgue norm estimates in $\mathbb{R}^3$

I'm currently working on a project in which I have to establish some estimates for some global Sobolev and Lebesgue norms. We know that if we have a bounded domain $\Omega$, then for any $q \leq p^*$ ...
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3answers
46 views

Definition of a compact operator

Operator compactness is characterized by maps the send the unit ball to relatively compact sets. Does anyone have a good justification for why we call this property compactness? The best ...
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0answers
22 views

$K$ is convex.Is this true that ${K^ \circ } \ne \emptyset$? [on hold]

Let $K$ is convex.Is this true that ${K^ \circ } \ne \emptyset$?
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34 views

resource on integral operators

Can you please suggest for me a good resource on integral operators.These are the specific topics that I am looking for: Bounded linear operators in hilbert space. Compact operators Spectral theory ...
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1answer
38 views

Given a singular matrix, I am tring to find an invertible matrix… (Finite Dimensional Space)

In coordinates and in a finite-dimensional space, how would I prove that given any singular $n$x$n$ matrix $A$, any $\epsilon\gt0$ and any matrix norm $||.||$, there is an invertible $n$x$n$ matrix ...
2
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0answers
24 views

Sequence in Hilbert space such that the angle beetwen any two of them is obtuse [on hold]

Let $\mathcal H$ be an infinite dimensional Hilbert space. Does there exists a sequence $(g_n)_{n\in\mathbb{N}} ,$ $g_n\in \mathcal{H}$ such that $$\left< g_i ,g_j \right> <0$$ for $i\neq j$? ...
3
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1answer
77 views

I am trying to find a Banach Space X and a singular operator (Infinite Dimensional Space)

I am trying to find a Banach space $X$ (Infinite dimensional Space) and a singular operator $A\in \mathcal L(X) $ such that for some $\epsilon \gt 0, $ there is no bounded linear operator $B$ with ...
2
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0answers
34 views

Hamiltonian: Commutator (II)

Problem Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ and an operator: $$A:\mathcal{D}(A)\to\mathcal{H}:\quad A=A^*$$ Denote for ...
4
votes
1answer
47 views

How would I show a functional is linear and bounded?

Using the following results, for any $f \in H^1(a,b)$, $f$ is continuous on $[a,b]$, and therefore, $$ \int_a^b f(x) dx = f(\zeta)$$ for some $\zeta \in (a,b)$. In addition $$f(c) = f(\zeta) + ...
3
votes
1answer
31 views

Surjective unbounded linear operator in Banach Spaces

Open mapping theorem says that bounded linear operator $T: X \to Y$ is an open mapping if $X$ and $Y$ are both Banach and $T$ is surjective. I am wondering what about unbounded linear operators? I ...