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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Find the codimension of $\overline{\operatorname{span}}\{S^n(1,4,-1,0,0,\ldots):n=0,1,2,\ldots\}$ in $\ell_2$.

Find the codimension of $A=\overline{\operatorname{span}}\{S^n(1,4,-1,0,0,\ldots):n=0,1,2,\ldots\}$ in $l_2$ where $S$ is the shifting operator to the right: $Se_i=e_{i+1}$. I don't quote understand ...
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1answer
25 views

Examples of bounded linear operators with range not closed

I've been trying to get some intuition on what it means for a bounded linear operator to have closed range. Can anyone give some simple examples of such an operator that does not have closed range? ...
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21 views

properties of vector space

Let $E$ be Banach space and $0<r<1$, $1\leq p<\infty$. Define the set $A$ as follow $$A=\left\{(x_j)_{j\in\mathbb{N}}\subset E :\sum\limits_{j=1}^{\infty}\left[\left|\left\langle ...
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1answer
24 views

Weak operator topology convergence of hermitian operators

Let $\{A_i\}$ be a net of hermitian operators on a separable Hibert space $\mathbb{H}$ and suppose that there is a hermitian operator T such that $A_{i}\le T$ for all i. If $\{<A_i h,h>\}$ is an ...
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15 views

Is the distance attained?

Suppose that we consider the set $K:=\{ x \in \mathbb{R}^n: \sum_{j=1}^n |x_j|^p \leq 1 \}$ where $0<p<1$. In this case the set isn't convex. Indeed, if we pick for example $x=(1,0,0, \dots), ...
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2answers
35 views

Prove equality of norms of operators

Let $e_i$=${\{\delta_{k,i}}\}_{k\ge1}$ $\in$ $l_2$, $i\ge1$, $A_n$ and $B_n$ - operators that are defined like this: $A_n\{x_i\}_{i\ge1}$ = $x_ne_1$, $B_n\{x_i\}_{i\ge1}$ = $x_1e_n$ ...
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37 views

Reference request: about inverse Laplacian operator

I am currently studying some problems about inverse Laplacian and the Yosida approximation and wishing to learn more about it. Here is a post about one of the problems that I am interested in. ...
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1answer
21 views

property of the Gelfand transform: why does an isometry map closed sets to closed sets?

The following is a theorem about self-adjoint subalgebra of $C(X)$ where $X$ is compact Hausdorff and the first half of its proof: Here are my questions: Why $\Gamma:\mathfrak{U}\to ...
2
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1answer
51 views

Closure of an Operator in $l^2$

Let $l^2$ denote the Hilbert space of all complex sequences $\phi = (\phi_j)_{j=0}^{\infty}$ such that $\sum_{j=0}^{\infty} |\phi_j|^2 < \infty$. Consider the linear subspace of $l^2$ defined by ...
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1answer
24 views

Determining isomorphism between sequence spaces

What's a good way to dis/prove that the $\mathbb{R}$-vector space of real convergent sequences and that of all real sequences are (linearly) isomorphic? The former space is isomorphic to $c_0$. Maybe ...
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1answer
20 views

continuity of a function and net convergence

The following is a statement and its proof in the Banach Algebra Techniques for Operator Theory by Douglas: I don't understand the last part of the proof. In order to show that $f$ is continuous, ...
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1answer
37 views

Show that for any function $f_0 \in L^p(E)$ there is a function $g_0$ in $C \subset L^p(E)$ closed convex

Let $E$ be a measurable set $1< p<\infty$ and $C$ a closed bounded convex subset of $L^p(E)$. Show that for any function $f_0 \in L^p(E)$ there is a function $g_0$ in $C$ for which ...
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0answers
21 views

Does this transformation(harmonic analysis) exist?

Assuming that there are two disjoint sets(A and B) of high dimensional $N^d$ integers. Each point $V$ can be expressed by a periodic function: $f(t)=v_1*cos(t) + v_2*sin(2t) + ...
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34 views

The inverse of Laplacian for different orders.

This question is related to my previous question here Let $u,v\in C_c^\infty(\Omega)$ and $\Omega\subset \mathbb R^N$ is open bounded with smooth boundary. Let $\Delta$ denote the Laplacian operator, ...
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1answer
21 views

Form of elements in closed linear span

Let $H$ be a Hilbert space, and $\{x_j\}$ an orthonormal set in $H$. Let $C$ be the closed linear span of $\{x_j\}$. I am trying to prove the following: Let $x$ be an element in $C$. Then $x=\sum ...
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0answers
15 views

Normal positive functional on Von Neumann algebras

Let $A$ be Von Neumann algebra. A positive linear functional ‎$‎‎‎\varphi‎$ on $A$ ‎is ‎said ‎to ‎be ‎normal ‎if ‎for ‎any ‎self‎adjoint and increasing nets such that ‎$‎‎u_{\alpha‎}\longrightarrow ...
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1answer
34 views

Given a $C_c^∞(G)$-valued random variable, is $C_c^∞(G)∋φ↦\text E[\langle\xi,φ\rangle]$ an element of the dual space of $C_c^∞(G)$?

Let $G\subseteq\mathbb R^d$ and $$\mathcal D:=C_c^\infty(G)$$ be equipped with some topology $\tau$ $\mathcal D'$ be the dual space of $\mathcal D$ and $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the ...
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2answers
113 views

Continuity of a function on a normed space

If $\mathbb{X}$ and $\mathbb{Y}$ are normed spaces, and E is a subset of $\mathbb{X}$ such that $f : E \rightarrow \mathbb{Y}$. How can I show that f is continuous if and only if for every closed ...
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2answers
45 views

Adjoint of an Operator in $l^2$

Let $l^2$ be the Hilbert space of all complex sequences $\phi =(\phi_j)_{j=0}^{\infty}$ such that $\sum_{j=0}^{\infty} |\phi_j |^2 < \infty$. Set $D= \{ \phi \in l^2 : \sum_{j=0}^{\infty} j ...
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1answer
23 views

What is a generalized stochastic process? I've found two different definitions. Are they equivalent?

Let $\mathcal D:=C_c^\infty(\mathbb R^d)$ and $\mathcal D'$ be the dual space of $\mathcal D$. What is a generalized stochastic process? I've found two different definitions in some textbooks: ...
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1answer
21 views

How to check compactness of linear operator

Let the continuous linear operator $T:l^2\to l^2$ be defined by $$T(x_1,x_2...)=(0,x_1,0,x_3,0,x_5,0...).$$ Are $T$ and $T^2$ compact? A linear operator $T: X\to Y$ is said to be compact if image ...
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17 views

$C^{\infty}_{C}(\Omega)$ it is dense in $ W^{1,2}(\Omega)=H^1(\Omega) $ ? I know that it is dense in $H^1_{0}(\Omega).$

$ C^{\infty}_{C}(\Omega) $ it is dense in$ W^{1,2}(\Omega) = H^1(\Omega)$ ? I know that it is dense in $H^1_{0}(\Omega) $ .
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0answers
14 views

Homotopy of bounded homomorphisms between Banach algebras

Let $A$ and $B$ be Banach algebras. Say that two bounded homomorphisms $\phi_0$ and $\phi_1$ from $A$ to $B$ are homotopic if there is a path $(\phi_t)_{t\in[0,1]}$ of bounded homomorphisms from $A$ ...
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1answer
47 views

Exercise 27 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein & Shakarchi's “Real Analysis” 2

I'm having trouble with the following problem: Prove that the operator $$Tf(x) = \frac{1}{\pi} \int_0^\infty \frac{f(y)}{x+y} dy$$ is bounded on $L^2(0,\infty)$ with norm $||T|| \leq 1$. I have ...
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0answers
36 views

embeddings of TVS are continuous

How to show that the two embeddings of TVS are continuous: $C^{\infty}_c\subset S\subset C^{\infty}$? According to Wikipedia, the definition of "Continuously embedded" is that " one normed vector ...
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49 views

Find the smallest value of $n$ for $P_n(x)$ to approximate $f(x)$ within $10^{-5}$ on $[-0.25,0.25]$

Let $f(x)=\ln(0.5+x)$ and let $P_n(x)$ be the $n$th Taylor polynomial for $f(x)$ about $x_0=0$. Find the smallest value of $n$ for $P_n(x)$ to approximate $f(x)$ within $10^{-5}$ on $[-0.25,0.25]$. ...
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11 views

Closable Multiplication Operator

I have the operator $M:Dom (M)\subset L^2(\mathbb{R}^N)\rightarrow L^2(\mathbb{R}^N)$, $Mf(x)=m(x)f(x)$, where $m$ is a continuous function and $Dom(M)=\{f\in L^2(\mathbb{R}^N)| mf\in ...
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0answers
37 views

C*-algebras: Proofs on $C_0(X)$

I'm looking to prove the following but am stuck, please can you help me? $C_0(X)$ is isomorphic as a C*-algebra to $C_0(Y)$ if and only if X is homeomorphic to Y, where X and Y are locally compact ...
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2answers
86 views

Closure of a subset of a normed space is equal to the set of limits of sequences in $A$

I'm not sure how to show that the closure of a subset $A$ of a normed space $\mathbb{X}$ is equal to the set of limits of sequences in $A$. Could someone help me?
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2answers
77 views

Showing that the metric $d$ is a norm

Let $X$ be a vector space, and $d:X\times X \to \mathbb{R}$ is a metric on $X$. Also suppose that $d$ is invariant under translations, i.e. $d(x,y)=d(x+z,y+z)$ for all $x,y,z \in X$. Is $d(x,y)$ for ...
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2answers
22 views

functions acting as linear functionals on their dual space

Supposing $f\in L^p$, where p and q are conjugate exponents, what does it mean that "f is completely determined by its action as a linear functional on $L^q$"? (Quoting Folland's Real Analysis here). ...
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1answer
100 views

Completeness of a finite dimensional linear subspace of X

How can I show that a finite dimensional linear subspace F of an arbitrary normed space X is complete, hence closed?
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40 views

Study the variation of $\frac{\Gamma(m-n^\prime+1,n^\prime q)}{\Gamma(m-n+1,n q)}$ with $q$

We define function $g(q)$ as the following: $$\frac{\Gamma(m-n^\prime+1,n^\prime q)}{\Gamma(m-n+1,n q)},$$ where $n^\prime \ge n+1$, $n^\prime \le m$. We note that $q$, $m$, $n$ $n^\prime$ are all ...
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0answers
21 views

Check proof that the embedding of the unit ball $B\subset X$ into $X^{**}$ in weak-* dense

I have to prove the following theorem: Let $X$ be a (real) Banach space, and let $B$ denote its closed unit ball, and let $\tau (B)$ denote its canonical embedding into $B^{**}$, the closed unit ...
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1answer
21 views

A question concerning the triplet $V\subset H\subset V^*$

In Brezis' Functional Analysis book, p. 150, there is an exercise about the triplet $V\subset H\subset V^*$, where $(V,\|\cdot\|_{V})$ is a Banach space, $H$ is a Hilbert space with the scalar product ...
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0answers
15 views

liminal and postliminal $C^*$-algebras

A $‎‎C^*$‎‎‎-algebra ‎$‎‎A$ ‎is said to be ‎postliminal (liminal) ‎‎ if for every non-zero irreducible representation‎$‎‎(H,‎\varphi)$ of ‎$‎‎A$ ‎we have‎‎‎ ‎‎$‎‎K(H)‎\subseteq ‎\varphi‎(A)$ ...
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1answer
20 views

Show that the nonlinear integral equation has a unique solution.

Show that the nonlinear integral equation $$f(x) = \int_0^1 e^{-sx}\cos{(\alpha f(s))}ds,$$ $0\leq x\leq 1$, $0<\alpha<1$, has a unique solution. I originally thought was some form of Fredholm ...
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1answer
47 views

Computing the inverse of Laplacian operator.

I am considering the following equation: $$ f(t):=\int_\Omega[(I-t\Delta)^{-1}\Delta(I-t\Delta)^{-1}u]\cdot u\,dx $$ where $u\in C_c^\infty(\Omega)$ and $t\geq 0$ a real number. $I$ is the identity ...
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0answers
23 views

Regarding integral operators being contractions

I have two half-questions that tie into one another. Suppose $T$ is an operator on $C([0, 1])$ defined by $$(Tu)(t) = \int_0^t (u(x))^2dx.$$ Show that T is not a contraction on the closed unit ball ...
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1answer
33 views

Open sets in the unitary group $ U(\mathcal{H}) $ of a Hilbert space $ \mathcal{H} $.

Let $H$ be an infinite dimensional Hilbert space and let $(x_i)_1^\infty$ be an orthonormal basis for $H$. Consider $U(H)$ the unitary group of the continuous unitary operators on $H$. Equip $U(H)$ ...
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1answer
50 views

Exercise 32 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein & Shakarchi's “Real Analysis” 2

I have a question about the following problem in Stein and Shakarchi's book. Consider the operator $T:L^2([0,1]) \to L^2([0,1])$ defined by $$T(f)(t) = tf(t).$$ (a) Prove that $T$ is a bounded ...
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52 views

What is the example of $L^p$ space which is not a Hilbert Space except $p=2$.

I know that $L^p$-norm satisfy the parallelogram law for $p=2$. But when $p$ is not equal to $2$ then it does not satify the parallelogram law and $L^p$ space is not Hilbert Space. For this I need a ...
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1answer
39 views

Show that $(\int\left(\sum_k|f_k|\right)^p)^{1/p}\le\int(\sum_k|f_k|^p)^{1/p}$

How to show that $(\int\left(\sum_k|f_k|\right)^p)^{1/p}\le\int(\sum_k|f_k|^p)^{1/p}$, or is it not true in general ? $\{f_k\}\subset L^p, G_n=\sum_1^n|f_k|$, I think $|G_n|_p$ is the sum above ...
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1answer
40 views

A equal closure of B

Let $A,B\subset X$ be subspaces of a normed space. I have quite a general question: If I want to show that $A=\bar{B}$ where $\bar{B}$ is the closure of $B$, is it enough to show that first ...
3
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1answer
23 views

How to indicate the space of sesquilinear forms?

Is there a canonical way to indicate the space of sesquilinear forms? For bilinear forms I can use the (0,2) Tensor space, but since sesquilinear forms are not linear in one variable I don't know how ...
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0answers
13 views

Why can “sufficiently smooth” distributions of $C_c^\infty([0,\infty)\times G)$ be represented as functions of $t\in [0,\infty)$ and $x\in G$?

Let $G\subseteq\mathbb R^d$ be a bounded domain and $$\mathcal D:=C_c^\infty([0,\infty)\times G)\;.$$ Assuming that $\mathcal D$ is equipped with the usual locally convex topology, the space of ...
2
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1answer
78 views

$\nabla^2(\|\boldsymbol{x}-\boldsymbol{x}_0\|^{-1})=-4\pi\delta(\boldsymbol{x}-\boldsymbol{x}_0)$ with distributions defined on Schwartz space

I know, from recent enlightening answers received here, that, if we define the distribution represented by Dirac's $\delta$ on the space $K$ of test functions of class $C^\infty$ whose support is ...
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0answers
13 views

Trace-Operator is compact?

Assume $\Omega$ is a smooth, compact riemannian manifold with non-empty smooth boundary $\partial \Omega$. Let $T$ be the Trace-Operator $T \colon H^1(\Omega) \to L^2(\partial \Omega), \ f \mapsto ...
0
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1answer
36 views

Operator Norm: Why is this formulation so rarely seen?

There are many equivalent definitions for the norm $||T||$ of a linear operator $T:X\to Y$. Citing the Wikipedia and some books about functional analysis, the equivalence of definitions are often ...
1
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1answer
21 views

problem on bounded linear operators

let X be the space of bounded real sequences with sup norm.define a linear operator T:X$\to$X by $$T(x) = (x_1,\frac{x_2}{2},\frac{x_3}{3},......) \forall x=(x_1,x_2,....)\in X$$ then,which one of ...