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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8 views

Prove that following are true for $\phi \in C_c^{\infty}(\mathbb{R}), \phi \ne 0$

Fix a function $ \phi \in C_c^{\infty}(\mathbb{R}), \phi \ne 0$ and set $u_n(x)=\phi(x+n)$. Let $1 \le p \le \infty$. Then Check that $u_n$ is bounded in $W^{1,p}(\mathbb{R})$ Prove that there ...
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1answer
18 views

Spectral projections, additivity

Let $K$ be a positive operator on a Hilbert space $H$. $Q_1$ and $Q_2$ are projections such that $Q_1\perp Q_2$. Is $$ E^{Q_1K Q_1} (1,\infty) + E^{Q_2K Q_2} (1,\infty) =E^{Q_1K Q_1 +Q_2K Q_2} ...
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4 views

Double integral of log supermodular function

I'm trying to prove that $\int\int\pi(v,w)f_{1}(w)f_{1}(v)dwdv\int\int\pi(v,w)f_{2}(w)f_{2}(v)dwdv>(\int\int\pi(v,w)f_{1}(w)f_{2}(v)dwdv)^2$ if $\pi(v,w)$ is log-supermodular and $f_1$ and $f_2$ ...
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1answer
16 views

Compactness of a bounded operator $A: l_1 \to l_1$

Let $l_1$ denote the space of absolutely summable sequences and $B(l_1,l_1)$ denote the space of all bounded linear operators from $l_1$ to $l_1$. I am trying to solve the following question Let $A ...
6
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1answer
43 views

A possible norm on a subspace of $C^\infty([0,1])$?

My question is related to this one: Take the vector space of infinitely differentiable functions on $[0,1]$. The standard norm of $C^k([0,1])$ is just the $\ell^1$-norm of the vector $(\|f\|_\infty, ...
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10 views

Author appendix Functional Analysis by F. Riesz and Sz. Nagy? [on hold]

Who is the author of the appendix of the book Functional Analysis by F. Riesz and Sz. Nagy?
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20 views

Show that for the triples $V \subset H \subset V^{*}$, the following are true

Let $H$ be a Hilbert space equipped with scalar product $(,)$ and the corresponding norm $|.|$. Let $V \subset H$ be a linear subspace that is dense in $H$. Assume that $V$ is a Banach space for ...
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12 views

Nearest Toeplitz matrix

Consider I have an arbitrary $NXN$ Hermitian matrix $A$. I want to derive a "suitable" Toeplitz matrix from $A$. I understand that there may be several ways to get a Toeplitz matrix from $A$ so ...
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1answer
15 views

Uniqueness of a Hahn Banach extension

Let $M = \{(x_n) \in l_1: x_1-3x_2=0\}$ and $f: M \to \mathbb{R}$ given by $f((x_n)) = x_1$. Let $g: l_1 \to \mathbb{R}$ be the Hahn Banach extension of $f$. Show that $g$ is unique. I know that ...
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1answer
101 views

Can we characterize the probability generating function as a linear operator?

For a nonnegative integer-valued random variable $X$ with $\mathbb P(X=j)=p_j$, we define the probability generating function of (the distribution of) $X$ by $$P_X(s):=\mathbb E\left[s^X\right] = ...
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0answers
9 views

Fractional Sobolev space on an interval

Consider a fractional Sobolev space $H_p^t(I)$ defined on an interval $I\subset \mathbb{R}$. When $I=\mathbb{R}$ the space can be defined via Fourier transform. Is it possible to do it when ...
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1answer
20 views

Functions over a $C$ vector space with geometric importance. (How to find the basis?)

Searching through our suggested exercises of linear and abstract algebra for solving, I found the following exercise. The reason I am posting this, is that because we haven't went through complex ...
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1answer
24 views

Calculating spectrum

The question is: Let $H = L^2(-\pi,\pi)$ and $[Au](x) = (1+x^3)u(x)$. Determine $\sigma(A)$. I'm reviewing for a test, so don't be concerned with "overhelping." I can determine the point spectrum ...
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6 views

Sobolev spaces on non-compact manifolds — independence on charts

Are there some standard references where basic facts about fractional-order (or at least integer-order) Sobolev spaces on non-compact manifolds are treated? More precisely I would like to be able to ...
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2answers
91 views

What is $H_0^1$ space?

I'm reading a book and it says that the $H_0^1(\Omega)$ space is defined as "the completion of $C_0^\infty(\Omega)$ w.r.t the Sobolev norm $\| \cdot \|_1$, where $C_0^\infty(\Omega)$ is the space of ...
2
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1answer
23 views

Evaluation of an integral associated with integral kernel of resolvent of Laplacian

I came across evaluating the following sort of integral when I was considering the integral kernel for resolvent of Laplacian $(I-\Delta)^{-1}$: $$ ...
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7 views

What is the importance of Leray Schauder non linear alternative

I'm questioning the importance of the Leray-Schaulder alternative, isn't more simple to use directly the Schauder fixed point theorem. Especially that, to prove the alternative we use the fixed ...
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0answers
15 views

How to show Plancherel's Theorem for Fourier Transform implies $L^2$ Transform Convergence.

The Plancherel Theorem for the Fourier transform $\hat{f}(s)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(t)e^{-ist}dt$ on $\mathbb{R}$ states that $$ ...
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12 views

subset S is dense in the set of all pure states of a $C^*$-algebra with respect to the weak-$^*$ topology

Let $A$ be $C^*$-algebra and $P(A)$ the set of all states $f:A\to \mathbb{C}$ such that: for all positive $ g\in A^* $ with $g\le f$ there exists $t\in [0,1]$ such that $f=tg$. I.e. $P(A)$ is the set ...
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0answers
14 views

continuity of some operator

Let $p \geq 1$ and let $A: l^{p + 1} \to l^{p + 1}$ be a continuous linear operator. Suppose that $A(x) \in l^{p}$ for all $x \in l^{p + 1}$. Is it true that the map $l^p \ni x \mapsto A(x) \in l^p $ ...
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1answer
20 views

Strong convergence of Spectral Projection

Let $H$ be a Hilbert space and $B(H)$ be the space of all bounded linear operators on $H$. Assume that $\{A_n\in B(H)\}_n$ strongly converges to $A$. $E^{|A|}(1,\infty)$ is a spectral projection of ...
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1answer
23 views

monotonic decrising p norm

I have the following inequality I need to prove: $$\|x\|_q \leq \|x\|_p \leq n^{\frac{1}{p}-\frac{1}{q}}\|x\|_q$$ For the right inequality, how can i prove $\|x\|_p$ is monotonically decreasing ...
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0answers
14 views

Two definitions of the operator $\exp(x)$ in $L^2(\mathbb R)$

The operator $x$ acts on a dense subspace of $L^2(\mathbb R)$ and is not bounded. So if we define $\exp(x)$ via the power series $\sum_{n=0}^\infty \frac {x^n}{n!}$, convergence will not follow in the ...
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1answer
18 views

A question on Bounded Approximation property

Let $V$ be a Banach space and we say that $V$ has the $C$-BAP if there exists a net of bounded finite rank operators $T_\alpha$ in $B(V,V)$ and a constant $C$ such that $\|T_\alpha\| \leq C$ for each ...
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0answers
14 views

Definition of the convolution with tempered distributions and Schwartz function

If $x \in \mathbb{R}^n$, $\varphi \in \mathcal{S}(\mathbb{R}^n)$ and $u \in \mathcal{S}'(\mathbb{R}^n)$ we define $(u \ast \varphi)(x)=\langle \tau_x \widetilde{\varphi} , u \rangle$, where we place ...
2
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1answer
13 views

Spectrum of a polynomial operator?

Let us have $A: l^2 \to l^2, A \in B(l^2)$. $$A(\delta_n)=3 \delta_{n}+i \delta_{n+1}$$ What is the spectrum of $A$? My approach: We can write down $A$ in a better form: $$A=3I - iR$$, where $I$ ...
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1answer
19 views

Riesz representation theorem for $C(\mathbb{T})$

I'm dealing with the Riesz representation theorem to prove that the dual $C(\mathbb{T})^*$ is (isometric to) the space of complex Borel measures on $\mathbb{T}$. On the other hand I've read that the ...
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1answer
31 views

Show if $\lim_{n \to \infty} \lambda_n=0$ then $Tu=\sum^\infty_{n=1} \lambda_n \langle u,e_n \rangle e_n$ defines a compact operator.

Let $\{e_n\}$ be an orthonormal basis in a Hilbert space $H$ and let $\{\lambda_n\}$ be a sequence of numbers. Define the operator $$T:H \to H$$ by $$Tu=\sum^\infty_{n=1} \lambda_n \langle u,e_n ...
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1answer
12 views

Linear map having finite norm in image

So, I have $f:K^d \to X$, where K is a field, and X is given normed space, where $f$ is supposed to be linear. Now, book states that if $e_k$ is a member of a standard basis for $K^d$ then ...
2
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1answer
34 views

What is the difference between weak and strong convergence?

What is the difference between strong and weak convergence? I am reading "Introductory functional analysis" by Kreyszig and I dont appreciate the differences between the two. Definition of ...
2
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1answer
18 views

Is this an elements of the Sobolev-Space $W^{1,2}\left(0,T,H^1(\Omega),H^1(\Omega)'\right)$?

Definition Let $p\in \mathbb{Z}$, $a<b \in \mathbb{R}$, and $X$, $Y$ be real Hilbert spaces. We define $$ W^{1,p}(a,b,X,Y) := \left\{ \varphi \, \Big| \, \varphi \in L^p(a,b,X),\, \varphi' ...
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3answers
20 views

What role does the range of an operator play in showing the operator is compact?

To show an operator is compact I understand you have to show the operator is the limit of finite rank operators. However the proof I have doesn't do this. I have an operator $$k:C0[,\pi]) \to C([0, ...
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1answer
20 views

Equivalent ways to study perturbed abstract Cauchy problem

I am considering the following abstract Cauchy problem on Banach space $X$: \begin{cases} u'(t)=Au(t)+\big(f(t)+Bu(t)\big),&t\in[0,T],\\ u(0)=x_0, \end{cases} Suppose $A$ generates a $C_0$ ...
0
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1answer
27 views

Problem. Convergence. Banach space. Weak topology

Let E be a Banach space and let $(X_n)$ be a sequence such that $X_n \rightharpoonup x$ in the weak topology σ(E,E'). Set: $S_n=\frac{1}{n}\sum_{k=1}^n(-1)^kX_k$ Does $Sn \rightharpoonup x $ in the ...
1
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1answer
45 views

What is the norm of the dual space $H^1(\Omega)'$?

I am working on the Bidomain-Model which, during a time interval [0,T], describes the electrical behaviour of the myocardial muscle considered as $\Omega \subset \mathbb{R}^3$. This model has partial ...
7
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1answer
2k views

L1 convergence gives pointwise convergent subsequence

I have been reading Terry Tao's notes on Real Analysis and there's a part he just says, but does not really explain, so I am wondering if someone here would. The notes are ...
0
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1answer
27 views

Finding an eigenvalue

I got the following statement to prove: Let $A \in \mathbb R^{n \times n}$ be a (column-)stochastic matrix, i.e. $A \geq 0$ (which means $a_{ij} \geq 0$ $\forall i, j \in \{1, \dots ,n\}$) and ...
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2answers
492 views

continuity in the strong topology implies continuity in the weak one

I have to prove that if $T:(E,\|\cdot\|_E)\rightarrow (F,\|\cdot\|_F)$ is a continuous and linear operator, and $x_h\rightharpoonup x$ in $E$, than $Tx_h\rightharpoonup Tx$ in $F$. So we know that $T$ ...
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1answer
24 views

Explicit example of two non-isomorphic Hilbert spaces with the same algebraic dimension [duplicate]

I´m wondering if there exist a vector space A and inner products: $\langle\cdot{,}\cdot\rangle_1$ and $\langle\cdot{,}\cdot\rangle_2$, such that: $\big( A,\langle\cdot{,}\cdot\rangle_1 \big)$ and ...
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11 views

If intersection of $ker {x_j}^{*} \subset ker f$implies f can be represent. [on hold]

I proved that $\cap_{j=1}^n ker {x_j}^{*} \subset ker f$ where ${x_j}^{*}$ are weak topology. Then, Can we represent $f$ by $f=\sum_{j=1}^n \lambda_j{x_j}^{*}$?? Also, I want to show that ...
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1answer
14 views

Show the operator $S$ defined by $(Sa)_n=\bigg(\frac{3}{5}\bigg)^n a_n$ is bounded on $l^2$ and find the operator norm. Is $S$ is invertible?

Define the operator $S: l^2 \to l^2$ by $$(Sa)_n=\bigg(\frac{3}{5}\bigg)^n a_n$$ for all $n \in \mathbb{N}$ an $a_n \in \mathbb{R}$. This is how we show it is bounded. ...
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0answers
24 views

Show $\rho_j =\sum^\infty_{k=1} \alpha_{jk} \epsilon_k$ is a compact operator

Let $\alpha_{jk}$ be numbers for $j,k \geq 1$ such that $$\sum_{j=1}^\infty \sum_{k=1}^\infty |\alpha_{jk}|^2 <\infty$$ Define $T: l^2 \to l^2$ as follows. If $x=(\epsilon_j)$, $y=(\rho_j)$ and ...
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8 views

For $A: Y \rightarrow Z$, find $z \in Z$ s.t. $z \notin A(Y)$

Let $Y =C[0,1]$ be the space of real-valued continuous functions equipped with the supremum norm $\|·\|_∞$, and $Z = \ell^∞$. $Af = (f(2^{−k}))_{k≥1}$ defines a bounded linear operator $A:Y →Z$. ...
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9 views

Dual space is isometrically isomorphic

Let $\{X_i:i\in I\}$ be a collection of normed spaces. If $1\leq p < \infty$, show that the dual space of $\bigoplus_p X_i$ is isometrically isomorphic to $\bigoplus_q {X_i}^\ast$, where ...
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10 views

heat equation-calculate the temperature of an bar

let an bar of lenght 50 cm, and temperature on t=0 is 100 degree. The question is calculate the degree on the middle of the bar. So i try to write the heat equation: $\dfrac{\partial u}{\partial t}= ...
3
votes
2answers
491 views

Eigenvalues and Spectrum

In algebra, I learned that if $\lambda$ is an eigenvalue of a linear operator $T$, I can have \begin{equation} Tx = \lambda x \tag{1} \end{equation} for some $x\neq 0$, which is equivalent to $\lambda ...
0
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1answer
31 views

$C_{c}^{\infty}(\Omega)$ is dense in $L^{\infty}(\Omega)$ with respect to the topology $\sigma(L^{\infty},L^{1})$

Show that $C_{c}^{\infty}(\Omega)$ is dense in $L^{\infty}(\Omega)$ with respect to the topology $\sigma(L^{\infty},L^{1})$, where $\Omega$ is an open subset of $\mathbb{R^n}$. My try: Let ...
3
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2answers
450 views
+200

Analytic Vectors (Nelson's Theorem)

Is there a (simple) proof for Nelson's theorem that a symmetric operator is essentially selfadjoint if it contains a dense subset of analytic vectors?
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0answers
10 views

Proving a linear operator is compact: understanding the statement “norm limit of a sequence of finite rank operators”.

I am having serious trouble understanding the proof that an operator is compact. This is the original question I asked and the proof is included very helpfully in the answer. Show if $\lim_{n \to ...
0
votes
0answers
21 views

Banach-Mazur distance from finite-dimensional subspaces of $\ell_p$ to the Hilbert space

I am reading a paper http://www.math.tamu.edu/~johnson/TF3.4.pdf by Bill Johnson and Andrzej Szankowski and having trouble grasping why $d_n(Z_m) \leq d_n(\ell_{p_{m+1}} ) = n^{|p_{m+1}-2|}$ in the ...