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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29 views

how can we get Pythagoras from the parallelogram law

When using the definition and properties of the inner product, we get the parallelogram law: $||x+y||^2= \langle x+y, x+y\rangle= \langle x, x\rangle + \langle x, y\rangle +\langle y, x\rangle ...
1
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1answer
25 views

Evaluating norm of the operator

I have to calculate norm of the operator $\varphi : l_{1} \rightarrow \mathbb{C}$, where $$ \varphi( (x_n)_{n=1}^{\infty} ) = \sum_{n=1}^{\infty} (-4)^{-n} x_{2n}.$$ My attempt was as follow: Let ...
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0answers
16 views

Definition of equi-absolute continuity

Could someone provide (or point me to) a definition of equi-absolute continuity for functions defined on an open bounded subset $\Omega \subseteq\mathbb{R}^n$? I only managed to find a definition for ...
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0answers
24 views

Estimating a sum

i want to show the following: Assume that $\sum_{m\in\mathbb{N}}{|i+\lambda_m|^{-p}}<\infty$ (where $(\lambda_m)_m$ is a sequence of real numbers). I want to show that then also holds for each ...
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0answers
38 views

Question about convergence in $L^2$ (revisited)

Yesterday I asked the folowing question: Question about convergence in $L^2$ which was answered negatively with a counterexample. Here, I wonder if one can find the right set to look at: Assume we ...
4
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1answer
35 views

Is the following statement true on $L^0$ spaces?

Let $(\Omega,\mathcal{F},P)$ be a probability space. Let $X,Y\in L^0(\Omega;\mathbb{R})$ two random variables taking values in $\mathbb{R}$. Is it true that: $$\int_{A} f(X(\omega)) P(d\omega) = ...
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2answers
36 views

Inequality error possibly. How are two inequalities equal?

Notation: $\underline{x}\in \Bbb R^n,||\cdot||_p =\left(\sum \limits_{i=1}^n |\cdot|^p\right)^{\frac1p}$ $$||\underline{x}||_p\left( \sum \limits_{i=1}^n |x_i + ...
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0answers
23 views

Elliptic regulartiy for nonlinear elliptic equations

Here is the question: Let us consider the Schrodinger type equation $$ \left\{ \begin{aligned} &-\Delta u + u = |u|^{p-2}u \quad \text{in } \mathbb{R}^N \\ &u\in H^1(\mathbb{R}^N) \qquad ...
1
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1answer
41 views

Determine if the set of all $f \in C(\mathbb{R})$ such that $f(1/2)$ is a rational number is a subspace of $C(\mathbb{R})$

Let $C(\mathbb{R})$ denote the vector space over $\mathbb{R}$ of all continuous functions on $\mathbb{R}$. Determine if the set of all $f \in C(\mathbb{R})$ such that $f(1/2)$ is a rational number is ...
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2answers
32 views

Hölder inequality conditions for $L_p$ spaces?

The Hölder inequality is the statement that if $f,g$ are measurable functions then $$ \|fg \|_1 \le \|f\|_p \|g\|_q$$ if $p,q$ are such that ${1\over p}+ {1 \over q} =1$. But it's not clear to me ...
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1answer
19 views

About the self-adjoint extension of an operator.

Let $B$ be a selfadjoint extension of an operator $A$ on a Hilbert space $H$. Let $\varphi \in \ker(A^\ast-z_0)$. Then i want to show that $\varphi + (z- z_0)(B-z)^{-1} \varphi \in \ker(A^\ast-z)$. I ...
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0answers
27 views

Continuous homorphism from $(\mathbf{R},+)$ to group of invertible elements in Banach algebra is differentiable

Let $A$ be a Banach algebra with $1$ and $\varphi\colon \mathbf{R}\to A$ be continuous such that $\varphi(0)=1$ and $\varphi(x+y)=\varphi(x)\varphi(y)$ for each $x,y\in \mathbf{R}$. The claim is ...
1
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0answers
18 views

Getting the minimum of a mixed functional

I have a functional $T$ defined on the attached picture. The functional always gives non-negative values. So it has a non-negative infinum I'm trying to figure out whether this infinum is ...
1
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2answers
44 views

Suppose that $f$ is differentiable on $\mathbb{R}$ and $\lim_{x\to \infty}f'(x)=M$. Show that $\lim_{x\to \infty}f(x+1)-f(x)$ exists and find it.

I've been stuck on this question for a long time now and was wondering if anyone could show me how it's done. So far I have done the following: Since $\lim_{x\to \infty}f'(x)=M$ then $\forall \epsilon ...
0
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1answer
44 views

Prove that a function is decreasing

Let $\left(\,c_m\,\right)_{m \in \mathbb{N}}$ be some coefficients which are all positive natural, $c_0=1$, and they are increasing in $m$. Define $$ f(y) = \frac{\sum\limits_{m=0} c_m \, \, ( y ...
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2answers
44 views

Misunderstanding a result from functional analysis

While reading page 111 of this book I got confused as to what the authors were doing in their counterexample of why strong convergence doesn't imply uniform convergence. I summarise it below Let ...
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0answers
18 views

Matrix induced norm by vector norm defined via a non-square weighting matrix

Let $W$ be a full-rank $m \times n$ matrix with $n<m$, i.e. it has linearly independent columns Define the wieghted norm on $\mathbb R^n$ as $\|x\|_W=\|Wx\|_{\infty}$. Is there a formula for the ...
0
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2answers
32 views

Convergence of spectrum with multiplicity under norm convergence

This article by Joachim Weidmann claims that, if a sequence $A_n$ of bounded operators in a Hilbert space converges in norm topology, i.e., $\|A_n - A\| \rightarrow 0$, then "isolated eigenvalues ...
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0answers
28 views

Proving a result on a sum of convex functions

Let $P=(p_{1} ,...,p_{n})$ and $Q=(q_{1} ,...,q_{n} )$ be two vectors of probabilities (do not sum to 1 so are not each a distribution). Define a function $F: (0,1)\to(0,1)$ with parameters $P$ and ...
2
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1answer
49 views

Question about convergence in $L^2$

Assume we have a sequence of functions $\{f_n\}_{n\geq 0}\subset L^2([0,1])$ such that $f_n \rightarrow f$ in $L^2([0,1])$, i.e. $$\lim_{n\to \infty} \int_0^1 |f_n(x)-f(x)|^2dx =0.$$ Is it then true ...
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0answers
38 views

Schauder basis for $c_0$

So, I am trying to prove that $c_0$ has the dual space $\ell^1$ (I know this proof is out there). Except my professor told me that a Schauder basis for $c_0$ is $(e_k)$ where $ e_k = \delta_{j,k}$ ...
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0answers
22 views

Show trigonometric function are complete on $L^2[0,2\pi]$

The proof is in the book but I couldn't understand it. Will appreciate your help. My doubts are in blue. Proof: Suppose $f(\theta)$ is any continuous, $2\pi$ periodic function ...
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0answers
23 views

Disjoint simple functions dense in $L^2(I^2, R^2)$

Suppose $$S=\{f\in L^2(I^2,R^2)| \exists h_1(x), f_1(x,y)=h_1(x), a.e. (x,y)\in I^2, \\ \exists h_2(x), f_2(x,y)=h_2(y), a.e. (x,y)\in I^2\}$$ For any $f=(f_1,f_2)\in L^2(I^2, R^2)$ and any ...
2
votes
1answer
34 views

A Lemma about the operator space

The following lemma comes from the book "C*-algebras Finite-Dimensional Approximations" by N.P. Brown and N. Ozawa P379 Lemma 13.2.3 Let $X_{i}\in B(H_{i})$ (i=1,2) be unital operator subspaces ...
1
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0answers
34 views

Name of the metric: $d(f,g)=\max \limits_{a\leq x \leq b} |f(x)-g(x)|$

What is the name of the metric: $$d(f,g)=\max \limits_{a\leq x \leq b} |f(x)-g(x)|$$ Where $f,g\in X$ where $X$ is the space of all continuous functions. I can't find any documentation on this ...
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0answers
8 views

Determining spectral bounds variationally.

I'm learning C0-semigroup theory (mainly from Arendt et al. (vector-valued Laplace-transforms and Cauchy problems), Engel & Nagel (One par. semigroups for linear evolution eq.),Evans (partial ...
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0answers
7 views

Lower order perturbations of 2nd order differential operators

Consider the well-known Hormander's sum of squares $P = \sum_{j = 1}^m X_j^2$, where $X_j$ are vector fields on a compact manifold $M$ of dimension $n$. Also assume, as is usual to this theory, $m ...
0
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1answer
48 views

why is $f(xx^*)$ is defined and $xf(x^*x)=f(xx^*)x$?

Let $A$ be a $c^*$algebra, $x\in A$ and $f:\sigma(x^*x)\to\mathbb{C}$ continuous and $f(0)=0$ ($\sigma(x^*x)$ is the spectrum of $x^*x$). Why is $f(xx^*)$ is defined and $xf(x^*x)=f(xx^*)x$? It is ...
0
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1answer
27 views

How to find the Dual space

If i consider the following space $$L^p_{\theta}=\{u:\Omega\rightarrow \mathbb{R}~\text{ mesurable}, \int_{\Omega} ||x|^{\theta} u(x)|^p dx<\infty\}$$ where $\Omega\subset \mathbb{R}^n$ is an open ...
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0answers
34 views

Self-adjointness of $X^2$

Let $X$ be a vector field on a compact (may be even complete) Riemannian manifold without boundary. I am wondering if $X^2$ will be a self-adjoint operator on $L^2(M)$. Any hints would be appreciated. ...
-3
votes
1answer
24 views

Example of sequence function on $C[a,b]$ [closed]

Please give me three examples of sequence $f_n$ in $C[a,b]:= \{ f \colon [a,b] \to \mathbb R \mid f \text{ is continuous} \}$ such that $$ \int_a^b | f_n(x) - f(x)|\ dx \to 0$$ ($L_1$ norm) as $n\to ...
1
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2answers
34 views

Can I define a bounded sequence whose Banach limit is not unique?

Banach limit, as a non-constructive object, is not unique. The Banach limit for some sequences, say, convergent sequences, sequences satisfying $a_n = a_{n+m}$ for all $n$ and some $m$, the Banach ...
1
vote
1answer
32 views

Weak convergence and lim inf and lim sup of the sequence of norms

Assume $x_n$ is a sequence in a Banach space that converges weakly to $x$. Then we know that $\|x\| \leq \lim \inf \|x_n\|$. 1)But can we say that $\lim \inf \|x_n\| < \infty$ or is this in ...
0
votes
1answer
21 views

Projection theorem for nonclosed subspaces

Is there a substitute for the projection theorem for Hilbertspaces (if $M$ is a closed subspace of $H$ then $H = M \oplus M^\perp$) in the case that $M$ is a linear subspace of $H$ which is not ...
2
votes
2answers
57 views

Banach space it isn't Hilbert space [duplicate]

How can give me two or three examples about Banach spaces which it is not Hilbert spaces with proof ( I mean why it isn't Hilbert spaces ) ?
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0answers
32 views

Can someone explain this problem I am having with the proof of the Riesz-Fischer theorem

Here is the form of the theorem I have; Let $\{e_n\}_{n=1}^{\infty} \in H$ be an orthonormal set (H a Hilbert space with inner product $(.,.)$) and let $(a_n)_{n=1}^{\infty}$ be an arbitrary sequence ...
2
votes
1answer
21 views

Natural structure over a set of measurable functions

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. Let $U$ be the set of all measurable functions over $(\Omega, \mathcal{F}, \mathbb{P})$ - i.e. the elements of $U$ are all measurable ...
0
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1answer
28 views

$X$ is inner product space then its completion is Hilbert space?

I have trouble finding a way to prove that the completion of my innerproduct space $X$ is a Hilbert space. How do I know that the norm on the completion of $X$ is induced by an innerproduct? Thanks ...
2
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1answer
22 views

Are the solutions of a Sturm-Liouville equation entire in the spectral parameter?

In $[1]$ the following (paraphrased) claim is made: Let $q\in L^1_{loc}([0,\infty);\mathbb{R})$, and suppose $\varphi$ and $\theta$ solve the one-dimensional Schrödinger equation \begin{equation} ...
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0answers
15 views

Does Schatten-p (quasi-)norm satisfy the norm inequality for 0<p<1?

I'm reading the paper by ANGELIKA ROHDE AND ALEXANDRE B. TSYBAKOV, ESTIMATION OF HIGH-DIMENSIONAL LOW-RANK MATRICES. And in the paper, they provide an inequality of the Schatten-p (quasi-)norm, ...
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0answers
37 views

Prob. 2.7-10 in Kreyszig's Functional Analysis Book: Is my solution good enough for anciliary purposes?

With valuable help from the SE community, I've managed to come up with the following solution to Prob. 10 after Sec. 2.7 in Introductory Functional Analysis With Applications by Erwine Kreyszig. I ...
3
votes
1answer
46 views

Strongest topology makes unit ball compact

Let $X$ be a Banach space and $X^*$ be its dual. Let $\mathbb{B}^*$ be the closed unit ball in $X^*$. The Banach-Alaoglu Theorem asserts that $\mathbb{B}^*$ is compact in the topology $\sigma(X^*, ...
3
votes
1answer
45 views

When does $f_n(x) = a_n \times (1 - nx)$ converge uniformly?

The sequence of functions $\{f_n\}_n$ is defined on $[0,1]$ by: $$f_n(x) = a_n \times (1 - nx),\ {\rm\ if}\ x \in ]0,\frac{1}{n}],$$ and $f_n(x) = 0$ otherwise, where $(a_n)_n$ is a positive ...
1
vote
1answer
34 views

Showing a set is not norm bounded

Consider the set $K = \{x(n) : x(n) \in \ell^p, \sum |x(n)| < 1\}$ $(0 < p < 1)$. I have shown that this set is weakly bounded, but I am now asked to show it is not originally bounded. where ...
2
votes
1answer
27 views

$k^2 e^{ikx} \rightharpoonup 0$ in the Sense of Distributions

So I'm concerned showing $k^2 e^{ikx} \rightharpoonup 0$ in the sense of distributions or in other words for any $\phi \in C_c^{\infty}(\mathbb{R})$ we have for any $\epsilon > 0$ $$ \left\lvert ...
0
votes
1answer
28 views

Coanalytic families of Banach spaces

Is it true that if $G$ is a coanalytic family of separable Banach spaces, which is not Borel, and $H\subset G$ is not Borel, then H is coanalytic? This is something I have come across. I am reading a ...
0
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0answers
10 views

Selfadjoint Operators: Relative Boundedness

Problem Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard an operator: ...
-1
votes
1answer
43 views

When is the $L^{2}$ norm smaller than the $H^{-1}$ norm?

If $u\in L^{2}$ then we can define the functional: $$u(\phi)=\int \phi u $$ for all $\phi \in H^{1}_{o} $. which means that $u$ is a linear functional in $H^{-1}$. Now for any $f\in H^{-1}$ ...
1
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0answers
25 views

Self-adjoint operator and vertex conditions in quantum graphs

Let $\Gamma$ be a metric graph with finitely many edges. Consider the operator $H$ acting as $\frac{-d^2}{dx_e^2}$ on each edge $e$, with the domain consisting of functions that belong to $H^2(e)$ on ...
-1
votes
0answers
14 views

What are endomorphism, automorphism of an operator algebra (or C*-algebra)?

Are these definitions true? Let $A$ be an operator algebra. Thus: 1) $f:A \to A$ belong to $End(A)\ $ if $\ f\ $ is homomorphism $ \ $i.e. $\ $ $f(ab)=f(a)f(b)\ $ for each $a,b \in A$. 2) $f:A ...