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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show that $B(H,K)$ is complete relative to this metric.

If $A, B \in B(H,K)$ and $d(A,B)=||A-B||$ is a metric on $B(H,K)$ show that $B(H,K)$ is complete relative to this metric.
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1answer
24 views

Show that A has an unique extension to a bounded operator on H

if $\{e_1, e_2, \cdots \}$ is an orthonormal basis for Hilbert space $H$ and for each $n$ there is a vector $Ae_n$ in $H$ such that \begin{equation*} \sum ||Ae_n||<\infty . \end{equation*} Show ...
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9 views

Unitary element in a C*-algebra

Suppose $\Bbb T$ is the unit circle and $M$ is the C*-algebra of all $2\times 2$ complex matrices. Consider the C*-algebra $A: = C(\Bbb T, M)$. Let $E$ and $F$ be the projections in $A$ given by ...
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15 views

Showing $\sum |\hat{f}(n)| \leq C \cdot \int_{0}^{2\pi} |f(t)| \ dt$ [duplicate]

If $f \in L^{1}[0,2\pi]$ define $\hat{f}(n)$ for $n \in\mathbb{Z}$ by $$\hat{f}(n) = \frac{1}{2\pi} \int_{0}^{2\pi} f(t) \cdot (\cos(nt) -i\sin(nt)) \ dt$$ Suppose $M$ is a closed linear subspace of ...
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22 views

Weierstrass approximation.

Show that the algebra generated by the pair of functions $\{l,2\}$ is dense in the set of all even functions that are continuous on $[—1, 1]$ I only know the above algebra dense in continuous function ...
3
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0answers
13 views

Is the Banach space of continuous functions on a compact space with a coutable base separable?

Let $X$ be a compact Hausdorff space with a countable base. Let $C(X)$ be the Banach space of complex valued continuous functions on $X$. Is $C(X)$ separable, i.e. does it have a countable dense ...
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1answer
26 views

Fundamental solution and Green's function

I am currently dealing with Poisson's equation $- \Delta u= f $ on some open domain $U$ and $u =g$ on the boundary $\partial U.$ Now a fundamental solution is a solution to $- \Delta u(x) = ...
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1answer
26 views

What $\mathbb{C}I$ means?

I've come across this expression $$ \mathbb{C}I $$ while studying operators algebras. $C^*$-algebras and AF-algebras, concretely. In Kenneth R. Davidson's book $\boldsymbol{C^*}$**-algebras by ...
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6 views

Existence of Schauder base for given operator

Suppose $A: l_2 \rightarrow l_2$ is a finite-rank linear bounded operator of dimension $k$. Is it true that there exists a Schauder orthonormal base for which only first $k$ columns will be nonzero ...
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1answer
19 views

Sum of bounded operators converge in B(X,Y)

Let $X$ be a normed space and $Y$ a Banach space and let $(A_n)_n$ be a sequence of bounded operators from $X$ to $Y$, and let there exist a sequence of positive nombers $(c_n)_n$ so that ...
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34 views

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
21 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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7 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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10 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 ...
2
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0answers
9 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
46 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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2answers
23 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
28 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
58 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 ...
2
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1answer
32 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
52 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
17 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
51 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
42 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 ...
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3answers
60 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
17 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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0answers
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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16 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 $\sigma(T)^* = \sigma(T^*)$. $\sigma_c(T)^* = \sigma_c(T^*)$. ...
1
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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. ...
0
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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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55 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
64 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
23 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
43 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
votes
2answers
47 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
66 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 ...