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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$\frac{\varphi(x+h)-\varphi(x)}{h}\to\varphi'(x)$ uniformly as $h\to 0$?

Let $\varphi$ be a bounded, differentiable function on $\mathbb{R}$ such that $\varphi'$ is bounded and uniformly continuous on $\mathbb{R}$. We want to prove that ...
4
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1answer
121 views

$\sigma$-weak topology versus weak operator topology

The reference text for this question is: Pedersen, Analysis Now, GTM 118. The $\sigma$-weak topology on $B(H)$ (the bounded linear operators on a Hilbert space $H$) is the weak$^*$-topology on ...
2
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1answer
39 views

Asymptotic behaviour of Fourier transform: $|F[f]|=|\lambda^{-k}F[f^{(k)}]|$ for absolutely continuous $f$

I read in Kolmogorov-Fomin's (p. 429 here) that if function $f:\mathbb{R}\to\mathbb{C}$ is such that $f^{(k-1)}$ [the $(k-1)$-th order derivative] on any finite interval and if $f,...,f^{(k)}\in ...
6
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0answers
77 views

Donsker's Theorem for triangular arrays

Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given $\alpha>0$, does some sort of Donsker's Theorem hold for $\left(\frac{X_i}{n^{\alpha}}\right)_{i=1}^n$? ...
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1answer
46 views

The polar of a set: Importance and Applications

Given a duality $(X,Y)$, for any subset $A\subset X$, we can define the polar set $A^{\circ}\subset Y$. In this sense, the polar relates sets and sets in the dual space. Since cones are sets, we can ...
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2answers
36 views

Prove $d(x,y)=\sup _{n} \left| \sum _{k=1}^{n}(x_k-y_k)\right |$ is a metric

Let $\gamma$ be the set of convergent series.$$\gamma = \{x=(x_k), x_k \in \mathbb{R} : \sum x_k <\infty\}$$ Prove that $(\gamma , d)$ is a metric space, with $$d(x,y)=\sup _{n} \left| \sum ...
2
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2answers
62 views

Polar decomposition

Every $x\in B(H)$ has a representation such as $x= u|x|$ (by polar decomposition) where $u$ is a partial isometry. In a paper, the author claims that$$u = {\rm strong} - \lim_{\epsilon\to 0} ...
2
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1answer
74 views

Weak and weak* topology coincide for a non-reflexive space that is isomorpic to its dual?

There are Banach spaces which are isomorphic to their second dual but not reflexive (most famously, the James space). Now let $X$ be such a space and $X'$ be its dual space and let $\phi:X\to X''$ be ...
2
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1answer
23 views

Some closed subspace of $l_2$?

$(a)$ I was trying to define a continuous linear map $T$ on $l_2$ whose kernel would be the $A$ and can conclude $A=T^{-1}(0)$ and hence closed set? could anyone help me to solve any of one?
3
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84 views

Functional inequalities involving cubing and incrementing

Consider the set $S$ of positive increasing invertible functions $f$ satisfying: $$f((x+1)^3-1)≤(f(x)+1)^3-1$$ $$f(x^3)≥(f(x))^³$$ $$f(x)+1≤f(x+1)$$ for all positive real $x$. Clearly the identity ...
2
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1answer
51 views

Ultraweak closed left ideal of a von Neumann algebra

The following is a proposition of Takesaki's Operator Theory: My questions are: 1- He claims for two sided ideal $\cal m$, $e \in M\cap M'$. While I think for $\sigma -$ weakly closed two sided ...
2
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0answers
51 views

norm over differentiable functions computable from derivatives only

I'm running an optimization algorithm, minimizing the norm $||f-\hat f||$ of some objective function $f(x_1,x_2,x_3,y_1,y_2,y_3)$. The function $f$ cannot be computed directly, but its second ...
2
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1answer
52 views

Theorem 2.3-2 in _Introductory Functional Analysis With Applications_ by Erwine Kryszeg

Here's the statement of Theorem 2.3-2 in the book mentioned above: Let $(X,||\cdot||)$ be a normed space. Then there is a Banach space $\hat{X}$ and an isometry $A \colon X \to W$, where $W = A(X)$, ...
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2answers
40 views

The fixed point in Brouwer's Theorem need not be unique.

What does it mean for a fixed point to be unique? I'm thinking that it means that you can have multiple values of a fixed point. But, a fixed point is one where $f(x) = x$. So to have repeated ...
2
votes
1answer
69 views

Infinite direct sum of Hilbert spaces

Let $\{H_i\}_{i \in I}$ be an infinite collection of Hilbert spaces. I am trying to understand their "Hilbert space direct sum". $\bigoplus H_i$ (algebraic sum) is an inner product space in a ...
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2answers
50 views

Power Series: Derivative

Given a Banach space $E$. Consider a series: $$|t|\leq R:\quad\sum_{k=0}^\infty A_k t^k\quad(A_k\in E)$$ Is there an elegant proof of: $$\left(\sum_{k=0}^\infty A_k t^k\right)'=\sum_{k=0}^\infty A_k ...
1
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1answer
21 views

Operator which is invariant in terms of left shift

Let $l:l^{\infty}(\mathbb R)\rightarrow \mathbb R$ be a linear map( where $l^{\infty}$ is the set of bounded sequences) such that the following holds: (i) $l(Tx)=l(x)$, where T is the left shift ...
1
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1answer
79 views

Proving this set is dense in $\ell^2$

I found this weirdest question and was wondering how could this be proved. This question is a part of a beautiful semi-constructive built of two dense disjoint convex sets in $\ell^2$, which I find ...
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2answers
79 views

C*-Algebras: Contractive Morphism

Problem Given C*-algebras $\mathcal{A}$ and $\mathcal{B}$ with $\mathbb{1}_\mathcal{A}\in\mathcal{A}$. Consider an algebraic morphism $\pi:\mathcal{D}\subseteq\mathcal{A}\to\mathcal{B}$ with ...
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1answer
32 views

Proof (?) about weak contractions. Please check to see if I'm going about this correctly?

If $f:M \rightarrow M$ satisfies that $\forall x,y \in M$, if $x≠y$ then $d(f(x),f(y)) < d(x,y)$, then $f$ is a weak contraction. Is a weak contraction a contraction? I saw a counter example on ...
8
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150 views

Sampling theorem.

Let us consider \begin{equation} \hat{f}(x)=\sum_{n\in \mathbb Z}\left\langle\hat{f},e^{i n x}\right\rangle_{L^2[-\pi,\pi]} e^{i n x} \ \ \ \ \ \ \ \ (1) \end{equation} where $\langle g, ...
0
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2answers
40 views

Unbounded operator

Assume you have an operator $T : \operatorname{dom(T)}\rightarrow H$. Now we also know that $ran(T)$ is finite-dimensional. Does this imply that $T$ is bounded?( So is $T$ a bounded map $T \in ...
0
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1answer
22 views

Showing that $p(x)\mapsto p'(x)$ is not a continous linear transformation

I am trying to understand a result in Rynne & Youngson: Linear Functional Analysis. Regarding continous linear transformations, the following is stated: Let $X$ and $Y$ be normed linear ...
0
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1answer
33 views

Sobolev space $W^{1,2}((0,1))$ and boundary ODE - how does integration by parts goes?

As a part of a question about $W^{1,2}((0,1))$, I want to get a boundary ODE on $g$ and don't quite know how to integrate (?) in order to get the equation. let $g\in C^2[0,1]$ be our variable, $f\in ...
2
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1answer
101 views

The measure generated by the Cantor staircase and the intersection of the Cantor set with its translate

Suppose that $T$ is the shift $\bmod 1$ of the Cantor set by an irrational number $\alpha\in (0,1)$. Consider the measure $\mu$ on the interval $[0,1]$ generated by the Cantor staircase. I'd like to ...
1
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1answer
59 views

Understanding connectedness argument in proof of Analytic Fredholm Theorem

Let $X$ be a complex Banach space, and let $D \subset \mathbb{C}$ be a domain. Let $\mathcal{L}(X)$ denote the Banach space of bounded linear transformations $X \to X$. The Analytic Fredholm Theorem ...
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1answer
88 views

Tensor product of function space and vector space

In one book I found that they treat vector valued functions as tensor product of some function space and vector space. So for example $$ C(\mathbb{R},V) \simeq C(\mathbb{R})\otimes V \\ ...
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0answers
20 views

Equivalence of condition on function

In a book I am studying it states that a condition on a function $g$ as follows: Given the function $g: \Omega \times \mathbb{R} \mapsto \mathbb{R}$ is a Caratheodory function satisfying $$\sup_{|u| ...
0
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1answer
24 views

Necessary condition of optimality for functionals

Let $C(a, b)$ denote the set of all surjective and continuously differentiable functions $\alpha:[a, b] \rightarrow [a, b]$. Consider the functional on $C(a, b)$ $$ F[\alpha(t)] = \int_a^b ...
5
votes
1answer
114 views

orthonormal sequence in $L^2[0,1]$ - how to prove these following equivalent terms?

I've been asked this following very interesting question and would like some help figuring out why it is true :) Let $u_n$ be an orthonormal sequence in $L^2[0,1]$ Prove that the following are ...
5
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2answers
98 views

Example (?) of a Banach space containing an uncomplemented copy of itself

I was wondering the following: Background Question: Does there exist a Banach space $X$ which contains a copy $X_0 \subset X$ of itself that is not complemented? By "$X_0$ is a copy of $X$", I ...
2
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2answers
41 views

Range conditions on a linear operator

While reading though some engineering literature, I came across some logic that I found a bit strange. Mathematically, the statement might look something like this: I have a linear operator ...
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0answers
44 views

What is the Vapnik-Chervonenkis dimension of sigmoidal functions?

Consider the following class of functions: $F=\{f_w:R^d \rightarrow [a,b], f_w(x)=\sigma(w^Tx), \forall x\in R^d\}$, where $\sigma(\cdot)$ is a sigmoidal function (e.g. tanh, or sigmoid so it has ...
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1answer
20 views

How to see injection and boundedness

Lemma. If $A$ is a bounded linear operator defined on a Hilbert space and $\|Af\| \geq c\|f\|$ and $\|A^*f\| \geq c\|f\|$ for some constant $c$. Then $A$ has a bounded inverse. In the proof of ...
1
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1answer
118 views

Computing a Projection Valued Measure

I've recently begun learning about Projection Valued Measure and I'm a little confused. I understand that a Projection Valued Measure is a family of orthogonal projections $P(\Lambda)$ indexed by the ...
0
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1answer
38 views

Surjective Operators

Let $X$ be a separable Banach space. Show that there is a surjective linear operator $P: L^1(\mathbb N) \to X$ and a subspace $S \subset L^1(\mathbb N)$ such that $$\|P(x)\| = \inf_{y \in S} ...
3
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1answer
32 views

How to see this is an isometry

Let $X$ be a separable Banach space, and $(f_n)$ be a countable dense subset. Recall that for each $f_n$ there exists a linear functional $l_n \in X^*$ such that $\|l_n\| = 1$ and ...
2
votes
1answer
48 views

Show that a bounded linear transformation is continuous

I am not sure what this question is asking. A linear operator $T$ between normed spaces X and Y is bounded if and only if it is a continuous linear operator. But weak topology is not metrizable. I ...
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0answers
32 views

How to change the fundamental frequency of a sample signal?

So I am dealing with a 60Hz signal that is sampled at 1kHz. This 60Hz signal has many other harmonics (eg, 120 Hz, 180Hz..... and more). For some reason, we would like it to be 50Hz. Could we ...
2
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1answer
144 views

Examples of double dual spaces

I am looking for examples on double dual spaces. As I know $\ell_p $ is the double dual of $\ell_p$ for $1<p,q<\infty$, $\mathcal L_p $ is the double dual for $\mathcal L_p$ for ...
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2answers
35 views

Decomposition procedure of closed subspaces in Hilbert Space $\mathcal H$.

Let $\mathcal H$ be a Hilbert Space. I then have a theorem saying: Let $U$ be a closed subspace of $\mathcal H$. Then we can write $\mathcal H = U \oplus U^{\bot}$. Now why is the following ...
2
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1answer
54 views

Theorem 2.4-3 in Kryszeg's *Introductory Functional Analysis with Applications*

In the book, Introductory Functional Analysis with Applications by Erwin Kreyszig, Theorem 2.4-3 states that every finite dimensional subspace $Y$ of a normed space $X$ is closed in $X$. Now the ...
2
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1answer
38 views

continuously embedding

Let $X$ and $Y$ be two normed vector spaces, with norms $||·||_X$ and $||·||_Y$ respectively. If there are constants $C_1, C_2≥0$ such that $||·||_Y \leq C_1||·||^{1/2}_X+C_2||·||^2_X$ for every $x\in ...
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1answer
46 views

Isn't this just the definition of weakly convergence?

Hi I come across this question when reviewing my notes. But isn't this just the definition of weak convergence? Am I failing to understand the problem? The definition given on my notes is A ...
3
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1answer
63 views

Soft Question: Linear Algebra Textbook to serve as a good foundation for Functional Analysis?

I want recommendations for an advanced linear algebra textbook that focuses on theory and provides adept background to support an advanced undergraduate or beginning graduate course Functional ...
0
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1answer
45 views

Find a sequence in the unit ball of L^1([0,1]) that has no convergent subsequence in the weak topology.

This is a question I come across when reading my notes. Find a sequence in the unit ball of $L^1$([0,1]) that has no convergent subsequence in the weak topology. By a corollary of Banach-Alaoglu, ...
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1answer
30 views

Norm of Triangle Matrix

How to find the norm of the following matrix, please? Thank you! $$T := \begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix},$$ and $$\|T\| = \sqrt{n^2+1}.$$
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1answer
50 views

Find a linear operator such that $\|T\| > \sup_{\|f\| \leq 1} \left|\langle Tf, f \rangle\right|$

Find a linear operator $T: \mathbb C^2 \to \mathbb C^2$ such that $$\|T\| > \sup_{\|f\| \leq 1} \left|\langle Tf, f \rangle\right|,$$ where $\langle \cdot, \cdot \rangle$ is the standard inner ...
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0answers
26 views

Show that $\pi(M)'' = \pi(M'')$

Let $M$ is a $*-$ subalgebra of $B(H)$. Let $\bar H$ denote the direct sum $\sum H_i$ where $\{H_i\}$ is a family of replicas of $H$. Define $$\pi :x\in B(H) \to \bar x \in B(\bar ...
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0answers
36 views

How to show this space is NOT reflexive

Consider the Banach space $X$ of null sequence whose elements are complex sequence which converges to $0$. In addition the norm is defined as $$\|(a_1, \dots, a_n)\| := \sup_n |a_n|.$$ Show this ...