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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When does Gelfand Naimark Theorem Hold?

I was going through the proof of the Gelfand Naimark Theorem for the Unital Commutative Banach Algebras. In proving that each character has norm equal to $1$, we used the fact that $||e|| = 1$ where ...
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54 views

Measurability of integral

Consider a function $f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$ which is continuous in the first argument, measurable in the second. Let $m: \mathcal{B}(\mathbb{R}^m) \rightarrow ...
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67 views

Intuition concerning Schwartz kernels of Operators

Consider a (for example differential) operator $A$ acting on an appropriate function space over a smooth compact manifold without boundary. Using the Schwartz kernel $K(x,y)dy$ of the operator, its ...
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32 views

Clarification about the space $C^0 ([-T,T],B)$

Let $B$ be a Banach space (in particular, $B$ is a function space equipped with the supremum norm). The space $C^0 ([-T,T],B)$ is the set of continuous functions on $[-T,T]$, valued in $B$. ...
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132 views

Proof of the completion of a metric space using cantors diagonal argument and showing a diagonal sequence is cauchy

I am studying applied functional analysis out of Applied Analysis by John Hunter. In chp. 1 of the text it gives a proof for the completion of a metric space. I am having trouble with understanding ...
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154 views

a normed vector space is normed closed iff it is weakly closed.

The claim is A subspace of a normed vector space is normed closed iff it is weakly closed. I can show one direction. Strong convergence implies weak convergence, so it is weakly closed. But I have ...
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147 views

Scale invariance and the Mellin transform?

The Mellin transform of a function is given by: $\mathcal{M}[f](s) = \int_0^{\infty}x^{s-1}f(x)dx$ Supposedly, the magnitude of the Mellin transform is invariant to scaling, analogous to how the ...
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95 views

Transpose of the Hilbert-Schmidt operator

Let $X = L^2(\Omega)$, $\Omega \subset \mathbb{R}^N$ be an open set (or a $\sigma$-finite measure space), $B \in L^2( \Omega \times \Omega)$. Then the Hilbert-Schmidt operator $T \in \mathcal L(X)$ ...
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109 views

Contradiction achieved with the Pettis Measurability Theorem?

$\bf{\text{(Pettis Measurability Theorem)}}$ Let $(\Omega,\Sigma,\mu)$ be a $\sigma$-finite measure. The following are equivalent for $f:\Omega\to X$. (i) $f$ is $\mu$-measurable. (ii) $f$ is ...
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128 views

Is exponentiation defined for non-self-adjoint operators?

In a book I am reading (Blank, Exner, Havlíček: Hilbert Space Operators in Quantum Mechanics), functions of operators are defined via spectral decomposition for self-adjoint operators. Spectral ...
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1answer
63 views

$B$ be an Infinite dimensional Banach Space and $T:B\to B$ be an continuos operator

Let $B$ be an Infinite dimensional Banach Space and $T:B\to B$ be an continuous operator such that $T(B)=B$ and $T(x)=0\Rightarrow x=0$ which of the following is correct? $T$ maps bounded sets into ...
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130 views

Complemented Banach spaces.

Let $X$ be Banach space and $Y$ a closed subspace of $X$. Assume that there exist a closed "subset" $Z$ of $X$ with the properties: $Z\cap Y=\{0\}$ and every $x\in X$ can be written in a unique ...
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174 views

an application of Hahn-Banach theorem

Let $M$ be a subspace of $L^1(\mu)$.Construct a bounded linear functional on $M$ such that there are two (hence infinite) different linear extensions preserving norm to $L^1(\mu)$. Someone outlined me ...
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40 views

bounded sets $ B\in X/M$ where $X$ is a normed vector space.

Let $X$ be a normed space, $ M\le X$ a linear subspace. Let $ X/M$ with the quotient norm. Prove that $ B \subset X/M$ is bounded iff there exist a bounded set $A\subset X$ such that $ B\subset [A]$. ...
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1answer
62 views

Problems with understanding the proof for existence of projections to a close convex set on a Hilbert space

In the setting of an introduction to functional analysis course, I have read the following statement: Let $H$ be a Hilbert space and let $A\subseteq H$ be a closed convex set. Then there exist a ...
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1answer
76 views

What is the standard (?) operator norm usually used in functional analysis?

I am studying introduction to functional analysis, in my lecture notes I have seen that a norm on functions is used in some proofs. For example I have seen the following: We note that for every ...
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1answer
69 views

$L^2$ dot product on surface of a sphere

If you have function $M,N:\mathbb{R}^3\rightarrow \mathbb{R}^3$ that are $M,N \in C^{\infty}$. Can we infer from this that on every surface of a sphere $B(0,R)$ this is a dot product: $$\langle f, ...
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388 views

proving that the quotient linear map of a continuous linear map is also continuous (normed spaces)

Let $X,Y$ be a normed vector spaces over $\mathbb k $, $T:X\to Y$ a $\mathbb k$-linear continuous map ($\mathbb k$ could be $\mathbb R$ or $\mathbb C$). Let's consider $ \hat T: X/Ker T \to Y$ the ...
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1answer
140 views

$S=\{x\in \ell^2 : \|x\|<1\}$ is compact?

$\ell^2$ denotes the set of all real square summable sequence with standard $\ell^2$ norm, $S=\{x\in \ell^2: \|x\|<1\}$ is compact? interior of $S$ is compact?closure of $S$ is compact? $\ell^2$ ...
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54 views

characterization of an infinite matrix mapping and continuity

Show that an infinite matrix mapping $A=[a_{ij}]$ $:l^{\infty}\to l^{\infty}$ is continuous iff $sup_{i\in \mathbb N}$ $ \sum_{k=1}^{\infty}{|a_{ij}|}=||A||<\infty$. Give a characterization of the ...
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1answer
29 views

proving that bs is banach

Let's define $B_s$ as the of real valued sequences $(x_n)$, such that $sup_{N\in \mathbb N} |\sum_{k=0}^{N}{x_k}| $ is bounded, and make it a vector space considering the usual pointwise operations ...
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95 views

How does showing that all norms are strongly equivalent imply that the identity map and the norm map are continuous functions?

In the begging of my introduction to functional analysis course the lecture started with a proof that all norms on $\mathbb{R}^{n}$ are strongly equivalent. Then the lecture said that from this we ...
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394 views

Measurability of supremum over measurable set

Consider a finite-valued function $f : \mathbb{R}^n \rightarrow \mathbb{R}$; and a closed-valued, measurable, set-valued mapping $S: \mathbb{R}^m \rightrightarrows \mathbb{R}^n$ . Measurability is ...
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54 views

positive definite operator and unique solution

In the page 32/39 of this paper http://archive.numdam.org/article/M2AN_1981__15_1_41_0.pdf they have the following equation: $T_h u_{h,t}+u_h=T_h f$ and say: "Since $T_h$ is definite positive (...) ...
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1answer
123 views

Completeness of Banach spaces

Let $X$ be a Banach space with norm $\|\cdot\|_X$. Define $$ \|f\|_p := \left (\sum_{n=1}^\infty \|f(n)\|_X^p \right )^{1/p} $$ for any function $f: \mathbb N \rightarrow X$. Let $\ell^p := \{f : ...
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208 views

A Riesz representation theorem without coercivity

Let $b:H \times H \to \mathbb{R}$ be a bounded bilinear form on Hilbert space $H$. Fix $u \in H$. Then $b(u, \cdot):H \to \mathbb{R}$ is bounded so $b(u,\cdot) \in H^*.$ Then $b(u,\cdot) = F_u(\cdot)$ ...
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66 views

Are periodic points dense in the unitary group?

In $U(1) = \{z \in \mathbb{C} : |z| = 1\}$, it is well known and easy to see that the set of $z$ so that $ z^n = 1 $ for some $n \in \mathbb{Z}_+$ are dense. Does this fact generalize to the group ...
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50 views

Showing that $e_k$ is an eigenvector

Lev $V$ be the space $L^2[-\pi, \pi] $ with the inner product $$\langle f,g\rangle=\int_{-\pi}^\pi f(x) \overline {g(x)} \, dx$$For integers $k$, let $e_k(t)=e^{ikt}$. Consider the operator $K$ on $V$ ...
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174 views

What are some characterizations of the strong and total variation topologies on measures?

The Wikipedia article on convergence of measures defines three kinds of convergence: total variation, strong, and weak. For weak convergence, a number of equivalent formulations are given, but not for ...
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42 views

what is the norm of this linear functional

$l:C[0,\pi]\to\mathbb{R}$ is defined by $l(f)=\int_{0}^{\pi} f(x)\sin^2(x) dx$ I need to find the norm of $l$, $|l|\le \int_{0}^{\pi}|f| dx$ as $|\sin^2 x|\le 1$ but I am not able to proceed ...
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2answers
142 views

Understanding why $L^{2}(\mathbb{R}^{n})$is incomplete

So I have just started a course which threw us right into the space that contains all continuous real valued functions. In other words, for $1 \leq p < \infty$, $$L^p(\mathbb{R}^n) = \{ f : ...
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1answer
104 views

Bounded Linear Functional

I have got stuck on this problem for quite some time now. Let $f$ be a measureable function that belongs to $X$, where $X$ is either $L^p$ ($1\leq p<\infty$) or $C_0$. Let $X_0=\{g \in X: ...
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121 views

Counterexample using counting measure

While proving that the norm of the mulplicative operator from $L^2(X) \to L^2(X)$ is the essential supremum of $|g|$ where $g \in L^\infty(X)$, I found that I need the $\sigma$-finiteness of the ...
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288 views

Linear combinations of delta measures

Let us consider the space of Borel, regular, complex measures on the real line, endowed with the total variation norm. Inside this space, I would like to characterize the space of all the finite ...
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1answer
87 views

Bounded Holomorphic Function - Banach Space?

Can someone, help me in this question, please? Let $U\subset\mathbb{C}$ be open set and $H_\infty(U)=\{f:U\to\mathbb{C}:f\text{ is bounded and homolorphic}\}$. Show that $H_\infty(U)$ is a closed ...
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164 views
2
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2answers
53 views

Whether these induced topologies are comparable?

$$\|x\|_1=\sum_{i=1}^{n} |x_i|$$ and $$\|x\|_2=(\sum_{i=1}^{n}|x_i|^2)^{1\over 2}$$ these two norm induce topologies on $\mathbb{R}^n$, I want to know whether they are comparable?
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86 views

Bound of $\log \det$

I want to find a bound to the function $$R(d_i, ...
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1answer
106 views

Finding Orthonormal Basis and minimum distance

Let $M$ be the subspace of $\mathbb{R}^4$ spanned by the vectors $$v_1=(1,0,0,0)$$ $$v_2 = (1,0,1,0)$$ $$v_3=(0,1,0,1)$$ A) Find a basis $M$ which is orthnormal with respect to the usual inner ...
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1answer
229 views

Is there a non-reflexive Banach space which is strictly convex?

I just come up with the fact that a space being strictly convex, does not implies it is reflexive (at least I never saw a proof of it). How can one construct a example of a non-reflexive Banach ...
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61 views

Does existence of weak spatial derivative imply existence of classical time derivative in this situation?

Let $f(x_1,...,x_n,t)$ be a function, where $(x_1,...,x_n) \in \mathbb{R}^n$ and $t \in [0,T].$ Denote by $f_{x_i}$ the weak (partial) derivative of $f$ wrt. $x_i.$ Is it possible for ...
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1answer
40 views

Relative compactness in $L^2(0,T,BD(\Omega))$

Here is the hypothesis of my problem : $T>0$, $\Omega$ is a bounded open subset of $\mathbb{R}^n$ with a regular boundary and I have a sequence of functions $(v_n)_n$ such that $$\forall n ...
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1answer
237 views

Convergence of functionals and weak convergence

I consider a Banach space $V$ with its dual $V'$. I had a sequence of functionals $\{f_k\}_{k\in \mathbb N} \subset V'$, and I wanted to show (strong or norm) convergence of $f_k \to f \in V'$. I ...
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1answer
118 views

How can I show that if $f\in L^p(a, b)$ then $\lim_{t\to 0^{+}}\int_{a}^b |f(x+t)-f(x)|^p\ dx=0$..

can anyone help me show that if $f\in L^p(a, b)$ then $$ \lim_{t\to 0^{+}}\int_{a}^b|f(x+t)-f(x)|^p\ dx=0.$$ Thanks, any help will be useful..
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313 views

Hamel bases without (too much) axiomatic set theory

Hamel bases have cropped up on the periphery of my mathematical interests a few times over my mathematical career, but I have never found the time or had a real need to look into them at any depth. ...
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1answer
41 views

Showing that an operator is $w^{*}-w$ continuous.

Let $X$ be a Banach space. Fix $x\in \ell_{1}[X]$, the space of all absolutely summable sequences in $X$. Define an operator $T:X^{*}\to\ell_{1}$, $\varphi\mapsto (\varphi(x_{n}))_{n=1}^{\infty}$. ...
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155 views

Many other solutions of the Cauchy's Functional Equation

By reading the Cauchy's Functional Equations on the Wiki, it is said that On the other hand, if no further conditions are imposed on f, then (assuming the axiom of choice) there are infinitely ...
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1answer
168 views

Exercise involving topological vector spaces, linear maps, and the quotient map

I'm doing a homework problem out of Rudin's $\textit{Functional Analysis}$ which is basically a proof of which I have completed some of it, but I'm not sure about the rest of it. Without further ado, ...
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1answer
118 views

What is the closure of the Laplacian on $L^2(0, \infty)$ with domain $D(\Delta):=C^\infty_0(0, \infty)$?

Let $\Delta$ be the operator on $L^2(0, \infty)$ defined as follows: $\Delta \phi:= \phi''$, with domain $D(\Delta):=C^\infty_0(0, \infty)$. Is $\Delta$ closed or closable? In the case, what is its ...
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2answers
430 views

Prove that sequence space $\ell_p(\mathbb R)$ is separable

Problem: Prove that metric space $\left \langle \ell_p(\mathbb R), d_p(x,y)=(\sum_{i=1}^{\infty} |x_i|^p)^\frac{1}{p} \right \rangle$ is separable. Where $\ell_p(\mathbb R)=\left \{ ...