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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31 views

Schur test and it's relation to representation theory

I was told by analyst who doesn't know about such things that Schur's test relating to boundedness of integral operators is somehow a version of Schur's lemma on irreducible representations, the group ...
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34 views

Tensor product space dense in $H_0^1$?

Given $C_c^{\infty}((a_i,b_i))$ for $i \in \{1,2\}.$ Is it then true that $$C_c^{\infty}(a_1,b_1) \otimes C_c^{\infty} (a_2,b_2)$$ is dense in $H_0^1((a_1,b_1) \times (a_2,b_2))$? Clearly, by ...
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1answer
15 views

Derivatives of pointwise converging sequence of functions may not converge anywhere?

On page 137 of Analysis by Lieb & Loss it is stated that The derivatives of a pointwise converging sequence of functions need not, in general, converge anywhere. I would have thought that ...
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14 views

Uniqueness of projection implies convexity [duplicate]

Prove that for a compact set A in finite dimensional Euclidean space X, A is convex if and only if for any point x in X, the projection of x to A is unique. If we know A is convex, we can show the ...
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1answer
25 views

Property of inner product on Hilbert space.

Let $H$ be a Hilbert space equipped with inner product $\left< \cdot , \cdot \right>$. Fix $u\in H$ and constant $R_0 > 0$. Define subset $K_u(R_0)$ of $H$ by $$K_u(R_0) =\{w \in H : \left&...
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1answer
43 views

Different approaches to differentiability in $L^2$

We can use different approaches to differentiability of $L^2(\mathbb{R})$ functions, e.g. we can say that $f\in L^2(\mathbb{R})$ is differentiable iff $f$ has a differentiable version (representative)....
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40 views

Baire measurable sets

I got the following setting: Let $X$ a uncountable set, $p \notin X$ and $X^* = X \cup \{p\}$. Let $$\mathcal O := \{O \subseteq X^* \mid O \subseteq X \text{ or } p \in O \text{ and } X \setminus O \...
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26 views

Is there a meaningful measure on analytic functions?

Let $\mathcal{B}$ be the functions analytic on the unit disk and continuous on its boundary. With the supremum norm this becomes a Banach space. Is there any way to define a meaningful measure on ...
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1answer
68 views

Proof verification: Something similar to Riesz-Fischer Theorem

Question: Suppose $\{f_n\}$ converges to $f$ in $L^p(\mathbb{R})$, $1\leq p<\infty$. Prove that there is a subsequence $\{f_{n_k}\}$ and $g\in L^p(\mathbb{R})$ so that $f_{n_k}\to f$ a.e. and $|f_{...
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1answer
37 views

What does the weak* topology on $\ell_2$ look like?

I am wondering about a way to construct a base or subbase for the weak* topology on $\ell_2$. I am fairly new to topology and functional analysis, so I apologize if the question is not precisely ...
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51 views

Existence of rotations between two points

Let $x,y\in\mathbb R^n$ ($n\in\mathbb N$) be two given points with the same Euclidean norm: $\|x\|=\|y\|$. Does there, in this case, exist an orthogonal matrix $U\in\mathbb R^{n\times n}$ such that $$...
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0answers
32 views

Constructing vector topologies (TVS's)

Consider the following theorem extracted from "An Introduction to Functional Analysis" by Charles Swartz (1992): Theorem 1: Let $X$ be a vector space. Let $\mathcal{U}$ be a family of subsets of $...
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0answers
34 views

Integral involving Dirac delta composition

I am trying to evaluate the integral $$\int dx \ \ f(g(x)) \ \delta(\alpha-g(x))$$ where $\alpha$ is a constant and $g$ some invertible function. Here's what I did: a change of variables $$g(x)=y$$...
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1answer
49 views

Proving continuity of a operator $T\colon E \to E'$ [duplicate]

Let be $E$ a Banach space over real numbers and $T\colon E \to E'$ linear such $T(x)(x)\geq 0$ for all $x\in E$, prove T is continuous. If $x_n\to x$ and $T(x_n)\to \phi\in E'$ then $T(x_n)(y)\to\...
3
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1answer
42 views

Why is a normal operator with certain spectral properties compact?

Given a compact operator, it is well-known that the Spectrum consists only of eigenvalues and possibly 0. Now I'm thinking about the inverse implication with additional conditions. So, given a normal ...
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2answers
32 views

Banach-Steinhaus problem

Let be $E$ a Banach space and $(x_j)_{j=1}^{\infty}$ a sequence in $E$ such for all $\phi\in E':~~\displaystyle\sum_{j=1}^\infty|\phi(x_j)|<\infty$. Prove that $$\displaystyle\sup_{||\phi||\leq1}\...
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70 views

Why are there no finitely additive measures on $\ell_\infty$ for which the measure of every ball is positive and finite?

As the question title suggests, why are there no finitely additive measures on $\ell_\infty$ for which the measure of every ball is positive and finite? Here, we do not assume that the measure is ...
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1answer
51 views

what is the reason of this of the following statement?

in a paper i saw the following statement: Let $\Phi:B(X)\longrightarrow B(X)$ is an additive and surjective map. If $T\in B(X)$ and for some $x\in X$ $Tx \otimes {\Phi(T)}^*f=\Phi(T)x\otimes T^*f$ ...
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0answers
27 views

Nonunital C*-Algebra: Proper Ideals

Given a C*-algebra without unit. Does there exist a nontrivial proper ideal that does not lie in a maximally nontrivial proper ideal? (For the unital case this follows easily by Zorn's lemma.) ...
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1answer
52 views

Prove there is a compact self adjoint $S:H\to H$ such that $S^3=T$.

Let $T:H\to H$ be compact and self adjoint. Prove there is a compact self adjoint $S:H\to H$ such that $S^3=T$. Is the $S^3$ means power of 3 or applying the operator 3 times? What is there to prove ...
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0answers
22 views

Prove for the family of operators $\{S(t)\}_{t>0}$ that $S'(0)$ exists

Let's consider a real-valued function $V\in L^\infty(\mathbb{R})$ and a family of operators $\{S(t)\}_{t>0}$ defined on $L^2(\mathbb{R})$ as follows: $$\qquad \qquad\left(S(t)f \right)(x) = \frac{...
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1answer
21 views

A question on uniform algebras

Let $A$ be a uniform algebra on a compact metric space $X$ Why the necessary condition for $A$ to be $C(X)$(the algebra of all complex-valued continuous functions on $X$) is that the maximal ideal ...
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1answer
29 views

Question on Inequality from Bartle's Elements of Integration: Riesz Fischer Theorem

I am puzzled how did Bartle get $$|g_k|\leq\sum_{j=k}^\infty |g_{j+1}-g_j|$$ (second last line)? I tried using Triangle Inequality and ended up with one extra term: $$\begin{align*} |g_k|&=|g_k-...
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1answer
46 views

the proof of variational principal for the principal eigenvalue (checking orthonormal subset)

Hi I am looking at part 3 of the proof in Evans Chapter 6. I have difficulty understanding "Furthermore from (6) and (7) we see that $(\lambda_k^{-1/2} w_k)$ is an orthonormal subset of $H_0^1(U)$. ...
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0answers
32 views

Sobolev Space with partial inner product

In my work, I encountered the following problem. Consider the set of real-valued functions, which are ``balanced'', that is the set of bounded functions $f(x)$ such that $\lim_{x\rightarrow \pm \...
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2answers
52 views

$||T-I|| < 1$ implies that $T$ is invertible.

Let $B$ be a banach space and $T : B \to B$ be a bounded linear transformation. If for identity transformation $I : B \to B$ , $||T-I||$<1 , then $T$ is invertible. || || is norm of ...
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0answers
43 views

Show that usual $L^2$ norm is equivalent to arbitrary norm $||\cdot||$ that satisfies 'convergence condition'

Given $L^2(\mathbb{R})$ consider a norm $||\cdot||$ on $L^2$ such that $(L^2,||\cdot||)$ is a Banach space and every $||\cdot||$-convergent sequence has a subsequence that converges almost everywhere. ...
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1answer
12 views

Is the delay line and bounded operator in a normed space?

Let $X$ be the normed space of all bounded real-valued functions on $\mathbb{R}$ with norm defined by $$ \|x\| = \sup_{t \in \mathbb{R}}{|x(t)|} $$ and let $T: X \rightarrow X$ be defined by $$ y(t)...
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2answers
82 views

Supremum of absolute value of the Fourier transform equals $1$, and it is attained exactly at $0$

Suppose that $f \in L^1(\mathbb{R}^n)$, $f \ge 0$, $\|f\|_{L^1} = 1$. How do I see that $\sup_{\xi\in\mathbb{R}^n} |\mathcal{F}(f)(\xi)| = 1$, and it is attained exactly at $0$?
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1answer
38 views

An exemple of strict inequality for reverse inequality Minkowski for space $L^p$, $0 < p <1$

Let be $0<p<1$. Suppose that we know that $$ \bigg(\int (u + v)^p\bigg)^{1/p} \geq \bigg(\int (u)^p\bigg)^{1/p} +\bigg(\int (v)^p\bigg)^{1/p}$$ for all $u,v \geq 0$ in $L^p$. I need find an ...
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2answers
23 views

Representation of functional on overlapping areas

I have given a functional $l$ on $C_c^\infty(\mathbb{R}^n)$. Now let's assume that for any $p \in \mathbb{R}^n$ we have a neighborhood $V_p$ and a $2\pi$-periodic $C^\infty$-function $u_p$ on $\mathbb{...
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1answer
44 views

Proving $\{T(x_n) \}$ converges to $0$.

Let $X$ and $Y$ be normed spaces and $T: X \rightarrow Y$ a linear operator. Prove that if the graph of $T$ is closed then $\{Tx_n \}$ converges to $0$ in $Y$, for all sequences $\{x_n \}$ that ...
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0answers
89 views

Continuous embeddings

Given the following exercise: We define $$C^1_c(\mathbb R_+) := \{f \in C^1(\mathbb R_+): \overline{\{x \in (\mathbb R_+) : f(x) \neq 0\}} \text{ compact in } (\mathbb R_+,|\cdot|)\}$$ (the ...
3
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1answer
31 views

About equivalent norms on a vector space

Definition. A norm $\|\cdot\|$ in a vector space $X$ is said to be equivalent to a norm $\|\cdot\|_0$ on $X$ if there are positive numbers $a$ and $b$ such that for all $x \in X$ we have $$ a\| x \|...
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1answer
44 views

If $\int_0^1 |f|dx= (\int_0^1|f|^p dx)^{1/p}$ for some $p > 1$, then $f$ is constant.

If $\int_0^1 |f|dx= (\int_0^1|f|^p dx)^{1/p}$ for some $p > 1$, then $f$ is constant. Since $\int _0 ^1 |f| dx = \int _\mathbb{R} fg dx $ where $g \mbox{ is characteristic function on } [0,1]$, I ...
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1answer
33 views

proving an element is unitary in a C* algebra

Let $p,q$ be projections in a unital C*-algebra $A$ and let $\tilde{A}$ be the unitization. I'd like to show that if $p\sim_u q$ (ie $q=zpz^*$ for $z$ unitary in $\tilde A$), then $q=upu^*$ for $u$ ...
7
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1answer
64 views

Prove that there exists a sequence $(x_n)$ such that $\sum_n a_n x_n$ diverges

So, here's a nice little result that I deduced using the closed graph theorem from functional analysis, but I'm wondering if there's a more elementary approach: Fact: Let $(a_n)$ be a sequence ...
3
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1answer
24 views

Proving linearity of an operator using boundedness.

I am considering an operator $K\colon \ell^2 \to \ell^2$ given by $$Kx = \sum_{n=1}^\infty e^{-n} \langle x , e_n\rangle e_n $$ where $e_n = (\delta_{k,n})_{k\in \mathrm{N}}$ is the standard basis on ...
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1answer
30 views

$\mathcal{l}^1$ is not complete for the norm $\|\cdot\|_\infty$

Let $\mathcal{l}^\infty = \{ (u_n) | u_n \in \mathbb{R}$ and $sup_{n \in \mathbb{N}}|u_n| < \infty \}$ and $\mathcal{l}^1 = \{ (u_n) | u_n \in \mathbb{R}$ and $\sum_{n=1}^{\infty} |u_n| < \infty ...
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0answers
58 views

A basis for $\text{span}_{L^2(0,1)} \left\{ \eta_\alpha,0<\alpha<1 \right\}$, and how compute coordinates

Let $$\text{span}_{L^2(0,1)} \left\{ \eta_\alpha,0<\alpha<1 \right\},$$ where we take $$\eta_\alpha(t)= \left\{ \frac{\alpha}{t} \right\} -\alpha \left\{ \frac{1}{t} \right\},$$ and $ \left\{ x ...
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2answers
61 views

Does existence of the second weak derivative of $f\in L^2$ imply existence of the first?

Let's consider a function $f\in L^2(\mathbb{R})$ for which the second weak derivative exists and lie in $L^2(\mathbb{R})$, i.e. there exists $f''\in L^2(\mathbb{R})$ such that for all $\varphi\in C_0^\...
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0answers
14 views

A comparison between absolute values of functionals

Let $A_0$ be a C*-subalgebra in a C*-algebra $A$. Let $\phi_0$ be a bounded linear functional on $A_0$ and assume $\phi$ is an extension of $\phi_0$ on $A$. I mean $\phi\in A^*$ with $\phi_{|_{A_0}}=\...
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1answer
31 views

Good, simple reference for Riesz-Fischer Theorem.

I am looking for a good, simple reference for the proof of Riesz-Fischer Theorem ($L^p$ spaces are complete). An example of a not so good reference in my opinion is Royden, where he uses "rapidly ...
6
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1answer
47 views

Norms on an Ultraproduct

Suppose $X$ is a Banach space and $\mathcal{U}$ is a non-principal ultrafilter on $\mathbb{N}$. I am interested in the Banach space $(X)_\mathcal{U}$, where we consider sequences $(x_i)_{i \in \mathbb{...
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1answer
36 views

variational principle for the principal eigenvalue

I am reading the proof of theorem 2 in chapter 6 Evan PDE. I have difficulty verifying the following part of the proof, i.e. 3 questions here. 1) The assumptions $u\in H_0^1(U)$ and $u\in L^2(U)=1$ ...
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2answers
35 views

Space of Lipschitz continuous functions is complete

Let $X$ be set of functions $f:[-1,1]\to \mathbb{C}$ such that $f(0)=0$ and there exists $\alpha>0$ such that $$ |f(t)-f(s)|\le \alpha |t-s| $$ for all $t,s\in [-1,1]$. Equip $X$ with the norm: $...
2
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0answers
39 views

Example of Hilbert space non isomorphic to $L2$

I'm looking for an example of a Hilbert space that can't be seen as the countable direct sum of $L^{2}(X,\mu)$ spaces nor subespaces of $L^{2}(X,\mu)$. Some idea to start? Thanks everyone.
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0answers
20 views

Non linear functional optimization under constraints

For some given positive functions $l(t)>0$ and $h(t)>0$, such that $h(t)>l(t)$, I want to solve this functional optimization problem on $a(t)$: $\min_a\int_0^T[l(t)\cdot\min(a(t),0) + h(t)\...
1
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0answers
120 views

Problem regarding continuous embeddings [duplicate]

Given the following exercise: We define $$C^1_c(\mathbb R_+) := \{f \in C^1(\mathbb R_+): \{x \in (\mathbb R_+) : f(x) \neq 0\} \text{ compact in } (\mathbb R_+,|\cdot|)\}$$ and for all $f \in C^...
0
votes
1answer
47 views

Spectrum of $T\in \mathcal{L}(E)$, such that $T^n=I$

Let $T:E \to E$ be a bounded linear operator, $E$ infinite dimensional Banach space, such that $T^n =I$, for $n\ge2.$ Show that $\sigma(T)\subset\{-1,1\}.$ My idea is show that $\|T\|=1$ initially,...