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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21 views

If $B$ is the closed unit ball of $X$, Why does $\varphi (B)$ is $\sigma $-dense in the closed unit ball of ${X^{**}}$?

Let $\varphi $ be the embedding of $X$ into ${X^{**}}$ . Let $\tau $ be the weak topology of $X$, and let $\sigma $ be the $weak^*$-topology of ${X^{**}}$--the one induced by $X^*$. If $B$ is the ...
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1answer
27 views

When the Multiplier algebra of a Banach algebra is exactly equal to the operator algebra?

Let A be a Banach algebra. B(A) and M(A) be the operator algebra and the multiplier algebra of A, respectively. When we have M(A)=B(A)?
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22 views

Banach space and Hamel Basis cardinality

No infinite-dimensional normed linear space with a Hamel basis having cardinality strictly less than c can be complete. Can we prove it without using AC or Hahn-Banach Theorem?
3
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0answers
35 views

Equivalence of norms in $C^1[0,1]$

i have the following problem/questions: I have to prove that $\lVert \cdot \rVert_1 \sim \lVert \cdot \rVert_{*} $ in $C^1[0,1]$; Where $\lVert \cdot \rVert_1$ is the usual $C^1[0,1]$ norm and ...
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0answers
28 views

Part of Lomonosov's Invariant Subspace Theorem

Let $X$ be a complex Banach space of infinite dimension, let $T\in\mathcal{B}(X)\backslash\{0\}$ be compact. Define $$\Gamma := \{S\in\mathcal{B}(X)\,|\,S\circ T=T\circ S\}$$and define, for each ...
2
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1answer
47 views

A Banach space of (Hamel) dimension $\kappa$ exists if and only if $\kappa^{\aleph_0}=\kappa$

A Banach space of (Hamel) dimension $\kappa$ exists if and only if $\kappa^{\aleph_0}=\kappa$. How will we prove the converse implication. One sided implication for Hilbert Space is proved in ...
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1answer
38 views

Basis for $l^{\infty}$

As the question stated, we know that $\{e_i\}$ doesn't form a basis for $l^{\infty}$. So how can we find a basis for $l^{\infty}$, no matter it is Schauder or Hamel basis.
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1answer
28 views

The Legendre-Fenchel transform of $BV$ semi-norm

I am reading a numerical paper in which it calls some "easy" facts from convex analysis but I can't justify it... Let $\Omega\subset \mathbb R^2$ be open. Define for a function $u\in L^1(\Omega)$ and ...
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1answer
39 views

existence of functionals

Let $X$ be a finite-dimensional normed space. Consider a non-empty convex set $C\subset X$ such that $0\notin C$. Notice that $C$ has a dense and countable subset $\{x_n\}$. $\forall n $ let $C_n= ...
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0answers
16 views

Gauss-Legendre quadrature error

I'm trying to evaluate the error in Gauss-Legendre quadrature formulae on $[a,b]$. So far I have that the error is less or equal to $$ \frac{f^{(2n)}(\xi)}{(2n)!}\langle p_n,p_n \rangle, \enspace ...
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1answer
29 views

If $S:(x_0,x_1,x_2,\ldots) \to (0,x_0,x_1,x_2,\ldots)$ Why does $0 \in \sigma (S)$? and what is $\dim(R(S))$ in $\ell^2$?

Consider the unilateral shift $S$ on $\ell^2$ defined by $$(x_0,x_1,x_2,\ldots) \to (0,x_0,x_1,x_2,\ldots)$$ Why dose $0$ is eigenvalue of $S$? and what is $\dim(R(S))$ in $\ell^2$?( Range of $S$)
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1answer
76 views

Hamel Basis in Infinite dimensional Banach Space without Baire Category Theorem

Prove that every Hamel basis in an infinite dimensional Banach Space is uncountable without using Baire Category Theory. We are assuming axiom that vector space dimension (if exist) is well-defined. ...
2
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0answers
21 views

Make mathematical sense of the Dirac well Potential Equation

A classical problem in quantum mechanics involving the Dirac Delta function is given by $$ y''+(\delta(x)-\lambda^2)y=0 $$ Then, to find ''bound states'', you solve on the right and find the ...
1
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1answer
26 views

$L^2(a_1,b_1;H_0^1(a_2,b_2))\subset L^2(a_1,b_1;L^2(a_2,b_2))$ and Convergence

Let $[a_1,b_1]\times[a_2,b_2]\subset\mathbb{R}^2$. Suppose $$u_n\rightharpoonup u\,\,\,\text{ weakly in } L^2(a_1,b_1;L^2(a_2,b_2))$$ and $$\{u_n\}\text{ is bounded in }L^2(a_1,b_1;H_0^1(a_2,b_2)).$$ ...
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1answer
17 views

Calculate the asymptotic growth of a sum that contains log or binom

I'm looking for a basic explanation how to calculate the asymptotic growth of sums. Take for example this one: $\sum_{i=1}^{lg(n!)} 2^{n^2}$ or this one: $\sum_{i=0}^{n} {n\choose{i}}$ The ...
2
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0answers
25 views

Eigen function of one Stochastic Process from the eigen function of another Stochastic Process

Let us consider a centred square integrable stochastic process $\{X_t:t\in [0,2]\}$. Also let the eigen values and the eigen function of the kernel of the covariance operator of $X_t$ are ...
2
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1answer
48 views

Can every vector space (over $\mathbb{R}$ or $\mathbb{C}$) can be a Banach space (or Hilbert space)?

For a vector space $V$ over $\mathbb{R}$ (or $\mathbb{C}$) with Hamel basis of cardinality $\kappa$ such that $\kappa^{\aleph_0} = \kappa$, can we define inner product(or norm) on $V$ such that $V$ is ...
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1answer
92 views

Is $T$ a compact mapping from $W_{0}^{1,2}\left(\Omega\right)$ into itself? [on hold]

Let $\Omega$ be an open bounded subset in $\mathbb{R}^{6}$ and $f$ be in $L^{8}\left(\Omega\right)$. For any $w$ in $W_{0}^{1,2}\left(\Omega\right)$, define $T\left(w\right)$ be in ...
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1answer
27 views

How to justify $\lVert \sum_{j=n+1}^\infty a_jh_j\rVert^2 \leq \sum_{j=n+1}^\infty a_j^2$ when $h_j$ are orthonormal

We work in a Hilbert space $H$. I want to show that a series $\sum_{j=1}^\infty a_jh_j$ converges where $h_j$ is an orthonormal basis of $H$. To do this, I want to show that the tail $$\lVert ...
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1answer
25 views

Hilbert space and uncountable cardinal

Given an uncountable cardinal does there exist Hilbert space with orthonormal basis of that cardinality?
5
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1answer
65 views

Equivalent formulations: pure contraction

I want to prove the following equivalence: let $T$ be a bounded self-adjoint operator on a Hilbert space $H$. TFAE: $\|Tx\|<\|x\|$ for each $x\in H\setminus\{0\}$ $\|T\|\leq1$ and ...
2
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1answer
38 views

Linear Operators Denseness and Injectivity

I'm studying for a Real Analysis prelim and have the following problem: "Let $X$ and $Y$ be normed spaces over $\mathbb{R}$ and let $$\mathcal{L}(X, Y) = \{T: X \rightarrow Y \mid T \text{ is bounded ...
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vote
2answers
23 views

Is there a relation eigenvectors and unitary operator.

I am trying the understand the spectral theorem as given in wikipedia link: https://en.wikipedia.org/wiki/Spectral_theorem I understand that eigenvectors are vectors and unitary operator is a ...
3
votes
1answer
38 views

Compact linear operator

Today in lecture we were told that for a linear compact operator $T$ on an infinite-dimensional Hilbert space with infinite-dimensional range, we have that $\ker(T)^{\perp}$ is infinite-dimensional, ...
1
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1answer
14 views

Range of operator always closed. Mistake in argument

Let $A \in L(X,Y)$ be a linear operator between Hilbert spaces and the operator $$\hat{A}: \ker(A)^{\perp} \rightarrow \operatorname{ran}(A)$$ is a restriction of $A$ which is bijective. Now ...
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0answers
19 views

$\sum\limits_{n=1}^\infty \sin{nx}= \frac{1}{2} \cot{\frac{x}{2}}$ in generalized functions D'

$D'$ are generalized functions (distributions) over D, which are finite (equal to zero outside some interval) functions in $C^\infty(\mathbb{R})$ I want to show that ...
1
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1answer
46 views

Borel Measures: Lusin

I'm trying to self-learn. Given the complex plane $\mathbb{C}$. Consider a Borel measure: $$\mu:\mathcal{B}(\mathbb{C})\to\mathbb{C}:\quad\mu\geq0$$ Regard a measurable: ...
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9 views

Calculate the horizontal plane of a box

How do I find the horizontal equatorial plane of a solid box (8x8" equal sides) if the box is set on one corner and the vertical plane passes through the opposite corner (straight up so the box would ...
2
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0answers
38 views

How to “uniformly” mollify a sequence without knowing the higher derivative.

The motivation and previous discussion of some related ideas can be seen here. My question: given a sequence $u_n\in u_0+H^1_0(\Omega)$ where $u_0\in H^1(\Omega)$ and $\Omega\subset \mathbb R^N$ is ...
3
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0answers
52 views

Looking for a bound on a function involving $\sinh$

Fix $T > 0$ and let $t \in (0,T)$ let $c > 1$ be a constant (which may be bigger than $T$). Consider the function $$f(c,t,T) = \frac{\sinh ((T-t)c)}{\sinh (Tc)}.$$ I am looking for a bound of ...
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0answers
71 views
+50

Integrating an infinite sum (statement from paper)

Let us work on a bounded domain $\Omega$ with Neumann BCs, with $\varphi_k$ and $\lambda_k$ being the orthonormalised eigenvectors and eigenvalues of the Neumann Laplacian. Define $$v(x,y) = \sum_{k ...
1
vote
1answer
28 views

How to get the basis of $L^2[0,1]$ from the basis of $L^2[0,2]$

Is there any way to derive orthonormal basis of $L^2[0,1]$ from the orthonormal basis of $L^2[0,2]$? Here $L^2[0,2]$: is space of square integrable centered stochastic process on $\Omega\times[0,2]$, ...
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1answer
33 views
+50

Prove or disprove that the following system $\left\{\frac{(-1)^{n-1}}{\pi}\left(\frac{\sin\pi t}{n-1-t}\right)\right\}_2^\infty$ is a Riesz basis.

Prove or disprove that the following system $$\left\{\frac{(-1)^{n-1}}{\pi}\left(\frac{\sin\pi t}{n-1-t}\right)\right\}_2^\infty$$ is a Riesz basis on $L^2(\mathbb R)$. I do not think it is a trivial ...
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0answers
23 views

Questions about stable rank of inductive limit of $C^\ast$-algebras

Let $A$ be an inductive limit of $\{A_n\}$ which are stable rank one. In Huaxin Lin's book An introduction to the classification of amenable $C^\ast$-algebra. The author assume that $\{A_n\}$ and $A$ ...
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0answers
34 views

Questions to the virial theorem

Let $H = H_0 + V$ be the Hamiltonian of the single electron where $H_0 = - \Delta, V = - \frac{\gamma}{|x|}$. Now one defines the dilation group $U(s) \psi(x) = e^{-ns/2} \psi(e^{-s}x), s \in \mathbb ...
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0answers
17 views

Convex functionals, min-max of linear functionals, etc

I was reading in this book on the well-known Courant-Fischer min-max Theorem, which states that if $\lambda_1(A)\geq \lambda_2(A)\geq \lambda_n(A)$ are the $n$ eigenvalues of a $n\times n$ Hermitian ...
2
votes
1answer
73 views

Isometry between $X^\ast/M^\perp$ and $M^\ast$.

Let $X$ be a normed linear space, $M \subset X$ be a subspace, $M^\perp = \{x^\ast \in X^\ast \mid x^\ast\big|_M = 0\}$ be the annihilator of $M$, $X^\ast$ the topological dual of $X$, and let's ...
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0answers
38 views

A question about discrete topological space [closed]

Let $(X,τ_δ)$ discrete topological space i have a question about it Is it locally compact space or not ?
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1answer
24 views

$u=\sum (u,\varphi_k)_{L^2}\varphi_k$ vs $u(x)=\sum (u,\varphi_k)_{L^2}\varphi_k(x)$ and related questions

I have some doubts after reading several threads. Let us work on a bounded domain $\Omega$ with Neumann BCs, with $\varphi_k$ and $\lambda_k$ being the orthonormalised eigenvectors and eigenvalues of ...
1
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1answer
32 views

Norm of multiplication operator in $\ell^2(\Bbb N)$ is $\|x\|_\infty$.

Fix a sequence $x = (x_n)_{n \geq 0}$. Then define $M_x: \ell^2(\Bbb N)\to \ell^2(\Bbb N)$ by: $$M_xy = (x_ny_n)_{n \geq 0},$$if $y = (y_n)_{n \geq 1}$. Supposedly we have $\|M_x\| = \|x\|_\infty$. On ...
1
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2answers
55 views

Density of smooth functions in $C([0,1])$

How can I show that smooth functions are dense in the space of continuous function on $[0,1]$? I know that we can use mollifiers. I wiki-ed it but can someone give me a rigorous proof?
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1answer
35 views

How to find the image of an arbitrary element under this operator?

Let $(e_n)$ be a total orthonormal sequence in a separable Hilbert space $H$ and define the right shift operator to be the linear operator $T \colon H \to H$ such that $T e_n = e_{n+1}$ for $n = 1, 2, ...
2
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0answers
44 views

If one expresses a function as a linear combination of other functions, can the linear combination relationship be inverted?

If one has a function say f, that can be expressed as a linear combination of another type of function say g, can one invert the relationship as a linear combination of the other function? i.e. if one ...
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1answer
18 views

Spectral Measures: Special Spectrum

Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
4
votes
1answer
37 views

For a self-adjoint $T$, if $T^k$ is compact, then so is $T$.

I'm studying functional analysis. Let $T:\mathcal{H}\rightarrow\mathcal{H}$ be a bounded self-adjoint linear operator on a Hilbert space $\mathcal{H}$. The problem is showing if $T^k$ is a compact ...
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1answer
12 views

Reducing Spaces: Decompostion

This thread is only Q&A! Problem Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ Regard a decomposition: ...
3
votes
2answers
41 views

Spanning set is closed.

Suppose $\{e_1,e_2,\ldots,e_n\}$ is an orthonormal set in $\mathscr{H}$ (Hilbert space) and define $$M \equiv \operatorname{span}\{e_1,e_2,\ldots,e_n\}.$$ Show that $M$ is closed. Can I show that ...
1
vote
1answer
25 views

Spectral Measures: Multi Version (III)

This question is only Q&A! Problem Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
0
votes
0answers
35 views

Formal decomposition of Hamiltonian into $A A^\ast$

Let $H = -\frac{d^2}{dx^2} + q$. Letting aside consideration of domains, I want to show that $H$ can be formally written as $H = A A^\ast$, where $A = -\frac{d}{dx} + \phi$ with some $\phi$ under the ...
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1answer
27 views

Borel Measures: Coproduct

I need this thread as lemma! (See the advice: SE: Q&A) Given Borel spaces $\Omega_\lambda$. Consider the coproduct: ...