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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3
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1answer
72 views

I need help understanding the proof of Lemma 2.4-1 from Kreyszig's Functional Analysis.

Lemma: Let $\{x_1, \ldots, x_n \}$ be a linearly independent set of vectors in a normed space $X$ (of any dimension). Then there is a number $c > 0$ such that for every choice of scalars $\alpha_1, ...
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0answers
58 views

How to prove the space $H$ is Banach?

Let $$H=\{\text{$f$:$f$ and its derivative are absolutely continuous and squared integrable in $\mathbb{R}$}\}.$$ The norm is $$\|f\|=\sqrt{\int_{-\infty}^{+\infty}|f(t)|^2\mathrm{d}t+\int_{-\infty}^{+...
5
votes
1answer
107 views

Categorical Banach space theory

Consider the category $\mathsf{NormVect}_1$ of normed vector spaces with short linear maps$^{\dagger}$ and the full subcategory $\mathsf{Ban}_1$ of Banach spaces with short linear maps. Both ...
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0answers
34 views

Weak convergence in $D^{1,p}(\mathbb{R}^n)$ and in $\mathcal{M}(\mathbb{R}^n)$

What it means: $$u_n\rightharpoonup u ~\text{weakly in }~ D^{1,p}(\mathbb{R}^n)\\ |\nabla(u_n-u)|^p\rightharpoonup\mu ~\text{weakly in}~ \mathcal{M}(\mathbb{R}^n)\\|u_n-u|^{p^{*}}\rightharpoonup\nu ~\...
4
votes
1answer
105 views

How can I prove that $f$ is inner product function

We know the polarization identity in inner product space : $$\langle x,y\rangle= \frac{1}{4} (\|x+y\|^2-\|x-y\|^2) + \frac{i}{4} (\|x+iy\|^2-\|x-iy\|^2) $$ But the question is if we have $(X,\|\cdot\|)...
0
votes
1answer
45 views

Obtain the triangle inequality from the Minkowski inequality

I want to prove: $$\|f-h\|_p \leq \|f-g\|_p + \|g-h\|_p$$ Minkowski inequality: $$\|f+g\|_p \leq \|f\|_p + \|g\|_p$$ Is $\|f-g\|_p \leq \|f\|_p + \|g\|_p$? It looks like we have: $$\left(\int_a^b |...
1
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0answers
50 views

$c_{00}$ is a dense subset of $c_0$

I would like to show that $c_{00}$ is a dense subset of $c_0$. I am not sure if I am overly simplifying the argument or even making the right argument for that matter. proof: Suppose that $x \in c_0$...
2
votes
1answer
35 views

For fn(z)= 1/nz, If we make fn(0)= 1, does that make the family of functions bounded?

I have a problem that requires me to use a theorem requiring a bounded family of functions. The family provided that I am supposed to use this theorem for is $f_n (z) = \frac 1 {nz}$ when $z \neq 0$ ...
9
votes
0answers
196 views

Energy inequalities with negative sobolev number.

Let $\phi\in H^{s}$ such that the following energy inequality is true: $$\|\phi(t,\cdot)\|_s \le\int^t_0 C \| P\phi(t,\cdot)\|_s \, dt $$ where $P$ is a strictly hyperbolic linear operator. For ...
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0answers
26 views

Lifting invertible elements in a $C^*$-algebra connected to the identity

Let $A$ and $B$ be unital $C^*$-algebras and suppose that there is a surjective *-homomorphism $f:A\rightarrow B$. Then any invertible element in $B$ that is connected to $1_B$ can be lifted to an ...
1
vote
2answers
35 views

Swapping series and linear operators

If $T$ is a continuous linear transformation between normed spaces. Under what conditions of $T$ and $(a_n)_n$ we have $T(\sum_{n=0}^\infty a_n)=\sum_{n=0}^\infty T(a_n)$?
2
votes
1answer
46 views

Gradient of the Fourier transform of a function

I am wondering if there is a simple way to express the first variation of the Fourier transform of a function as a function of said function. In other words, if $g:x\mapsto F(f)(x)$, where $F(f)$ is ...
3
votes
1answer
58 views

Lower semicontinuous energy functional on compact space of Lipschitz functions

Let $\Omega \subset \mathbb{R}^{n}$ be a bounded open subset containing $0$ and let $L>0$ be some positive constant. Consider the space $A_{0}=\{f \in C^{\infty}(\overline{\Omega}) \mid f \text{ ...
0
votes
1answer
42 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
vote
1answer
37 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 $|...
0
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0answers
43 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
votes
1answer
44 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) = \int_{...
0
votes
2answers
37 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 + y_i|^{(p-1)q}\right)^{\frac1q}+||\...
1
vote
1answer
52 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 ...
1
vote
2answers
62 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 ...
0
votes
1answer
30 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 ...
1
vote
0answers
21 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
vote
2answers
56 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
votes
1answer
51 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 \...
1
vote
2answers
48 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 $...
0
votes
2answers
93 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 $\...
2
votes
1answer
52 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 ...
2
votes
1answer
107 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}$ ...
2
votes
1answer
53 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
47 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 ...
0
votes
0answers
22 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
votes
1answer
87 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 $x^...
0
votes
1answer
30 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 ...
1
vote
0answers
35 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. ...
1
vote
2answers
86 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 ...
2
votes
1answer
275 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
32 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 closed?...
2
votes
2answers
147 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
42 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
24 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 ...
1
vote
1answer
108 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
votes
1answer
27 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} ...
4
votes
1answer
64 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 $\sigma(X^*, X)$....
3
votes
1answer
83 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
45 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
33 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
30 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 ...
-1
votes
1answer
73 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
vote
0answers
38 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 ...
4
votes
1answer
1k views

Integral of a square compared to the square of an integral

What can be said about a complex valued, continuous function $f$, defined on $[0,1]$, such that: $$ \int_0^1{|f|^2}=\left|\int_0^1{f}\right|^2 $$ I encountered this form as part of an exercise. ...