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 views

Strong approximation of operators.

If I want to approximate strongly an operator $T$ with another in a subset $A \in L(H)$ why is not enough to ask "for every $\epsilon>0$ and every $\eta \in H$ there is an operator $S\in A$ such ...
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0answers
5 views

If $\varphi f\in L^1(\mu)$ for every $f\in L^1(\mu),$ then $\varphi \in L^\infty$

Let $\varphi$ be a measurable function for which $\varphi f\in L^1(\mu)$ for every $f\in L^1(\mu).$ Show that $\varphi \in L^\infty(\mu).$
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4 views

Strong maximum principle for weak solutions?

For a general linear parabolic equation, is a strong maximum principle possible when the solutions are merely weak solutions (i.e. they lie in a Bochner space)? Is there some proof possible that does ...
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0answers
12 views

Question about the Averson's proof of the bicommutant theorem.

In the Averson's proof of the bicommutant theorem is proved that, if $A$ is a self-adjoint algebra of operators with trivial null space and $T \in A''$, for every $\epsilon>0$, $n=1,2..$ and every ...
2
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1answer
53 views

List of functions $f(cx) = C\cdot f(x)$

I was looking for some complex functions f(x), which satisfies the condition: $$\exists (c, C) \in \Bbb C^2 \backslash\{(1,1)\}, \forall x \in \Bbb C, f(cx) = C\cdot f(x)$$ Till now I have got ...
3
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0answers
26 views

$\sin(nx)$ does not contain Cauchy subsequence in $L^p([0,2\pi]) $ for $1\leq p < \infty$

$\sin(nx)$ does not contain Cauchy subsequence in $L^p([0,2\pi]) $ for $1\leq p < \infty$ My attempt: Set $f_n(x) = \sin(nx)$. Argue by contradiction, suppose there exists a Cauchy ...
9
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1answer
48 views

There is no continuous mapping from $L^1([0,1])$ onto $L^\infty([0,1])$

There is no continuous mapping from $L^1([0,1])$ onto $L^\infty([0,1])$. Proof: suppose $T:L^1 \rightarrow L^\infty$ continuous and onto. $L^1$ is separable, let $\{f_n\}$ be a countable dense ...
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1answer
23 views

Norm of functional associated to vector $p$-norm [duplicate]

I read that the norm of a linear functional $f:V\to K$, with $K=\mathbb{R}\lor K=\mathbb{C}$, associated to the $p$-norm $\|x\|=(\sum_{i=1}^n|x_i|^p)^{\frac{1}{p}}$, for $p>1$, is ...
3
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2answers
18 views

Question about a passage in the Bicommutant Theorem's proof.

In the Averson's book, in the proof of the Von Neumann's Bicommutant theorem there is this passage: ($A $ is a self-adjoint algebra of operators in $L(H)$) "Let $\xi_1$ be an element of the Hilbert ...
2
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2answers
25 views

Banach Spaces: Totally Bounded Subsets

As an easy consequence of Riesz' lemma it is known that infinite dimensional Banach spaces possess bounded subsets that fail to be totally bounded. On the other hand in finite dimensional Banach ...
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1answer
19 views

Lipschitz constants of projections

Consider two compact sets $A, B \subset \mathbb{R}^n$. Assume that the projection mappings $P_A: \mathbb{R}^n \rightarrow A$, $P_B : \mathbb{R}^n \rightarrow B$ have Lipschitz constant $1$ and $L$, ...
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1answer
14 views

Is there any difference between formally symmetric and formally self-adjoint differential operators?

I work with the well known book of Dunford/Schwartz "Linear Operators (Part II)". At first I should mention that the general difference between self-adjoint and symmetric operators is obvious to me. ...
3
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1answer
46 views

Sums of special vectors

Let $v$ be a vector obtained by taking a sum of $k$ vectors the of the form $(0,0,\ldots,0, -n, *,*,\ldots,*)$, where $"*"$ stands for either $0$ or $1$, and the position of the $-n$ entry can vary ...
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1answer
21 views

Question about total variation

When I was reading http://mathpost.la.asu.edu/~ylin/YLin_thesis.pdf , I didn't understand the following. Why is TV the sum of "jumps". It seems to me that (1.24) is a formula of arc length. But why ...
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1answer
26 views

Understanding the scalar product in the duality

I was trying to solve an exercise for my class, but then I have found somewhere a solution. I need to understand the meaning of a certain step. The exercise and the solution read as follow. Exercise ...
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1answer
49 views

Banach Spaces: Totally Bounded vs. Bounded

Are the finite dimensional Banach spaces precisely those ones in which subsets are totally bounded iff they're bounded?
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2answers
31 views

$L^{\infty} (X, \mu)$ is not separable? [on hold]

Show that $L^{\infty} (X,\mu)$ is not separable if $X$ contains sequences of disjoint sets of strictly positive measure?
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23 views

completion of a normed space [on hold]

Prove that if $M$ is a normed linear space,then there exists a unique(up to isomorphism)Banach space $X$ containing $M$ such that $\overline{M}=X$ Please help me.Thanks in advance.
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1answer
38 views

Continuous functional such that $f(x_0)\ne 0$

I read that in any locally convex topological space $X$, not necessarily a Hausdorff space but with linear operations continuous, for any $x_0\ne 0$ we can define a continuous linear functional ...
2
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1answer
20 views

How to prove the following map is a c.c.p map

Here is a quotation in a book "C*-algebrass and Finite-Dimensional Approximations" by Nate and Taka (P122). Let $A$ be a C*-algebra, $\Gamma$ be a discrete group and the $\alpha$ is an action of ...
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0answers
36 views

Darboux Integrable Functions vs. Uniform Closure of Simple Functions

Is there a Darboux integrable function $f:[0,1]\to E$ with values in a Banach space $E$ that is not the uniform limit of simple functions $s=\sum_\alpha\chi_{A_\alpha}$ taken over finite sums with ...
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1answer
62 views

Is $C(C(\mathbb R))$ notation for the set of continuous functions mapping $C(\mathbb R)$ to itself?

Given that in general functional analysis we have $C(\mathbb{R})$ being the set of all continuous functions, $f: \mathbb{R} \to \mathbb{R}$. However, could I use $C(C(\mathbb{R}))$ notationally to be ...
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1answer
17 views

Measurability of product of Borel measurable functions with different domains?

Suppose we are in the measure space $(\mathbb{R}, \Sigma(m), m)$ ($m$ is Lebesgue measure). Also, suppose $f, g \in L^{1}(dm)$. We define the convolution of $f$, $g$, by $(f * g)(y) = \int ...
2
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1answer
48 views

Topological and algebraic interiors

I read on a functional analysis book that in a normed, real or complex, space $V$ the algebraic interior of a set $S\subset V$ defined $J(S):=\{x\in S:\quad\forall y\in V\quad\exists ...
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0answers
13 views

$\big\| f_1(x) \overline{f_2(x)} |f_3 (x)|\big\|_{H^1(\mathbb R)} \le C_{>0}\|f(x)\|_{L^\infty(\mathbb R)}^2 \|f(x)\|_{H^1(\mathbb R)}$ holds? [duplicate]

I want to know that whether the following inequality holds or not for complex-valued functions $f_1$, $f_2$, $f_3$ on $\mathbb R$: $$ \big\| f_1(x) \overline{f_2(x)} |f_3 (x)|\big\|_{H^1(\mathbb R)} ...
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0answers
17 views

How can I prove that the negative biased triangular kernel is positive semidefinite

How can I prove that the following triangular kernel function defined in $[0, 1] \subset R^1$ $k(x, x') = (1 - 2|x-x'|)$ is a positive semidefinite function? It turns out to be psd function when ...
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0answers
32 views

Mean of Piecewise function resting on IID random variables

Suppose IID random variables $X_t \sim X$ with support on $[0,1]$ and continuous CDF $F(\cdot)$. I wish to compute the expected value (mean) of the a piecewise (continuous) function with form $$ \Phi ...
2
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0answers
54 views

A Tricky Weak Derivatives question

I recently came across the following statement and am having trouble proving it correct. I wonder if you could help. Given a weak derivative, $x'$, there exists an absolutely continuous ...
1
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1answer
23 views

Minima of convex Gateaux differentiable maps

I conjectured that: if $E$ is a reflexive Banach space and $F: E \to \mathbb{R}$ a convex Gateaux differentiable map (in other words all the directional derivatives $\frac{\partial F}{\partial ...
2
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0answers
63 views

$C(X)$ is finite dimensional iff $X$ is finite [duplicate]

If $X$ is compact Hausdorff space and $C(X)$ is the set of all continuous complex valued functions on $X$,then prove that $C(X)$ is finite dimensional if and only if $X$ is finite. My problem:If we ...
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0answers
21 views

Is a functional derivative a distribution?

I am just now learning about elementary distribution theory, and it seems that theory may bear on the topic of functional differentiation, which I've encountered in some books on quantum field theory. ...
1
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0answers
41 views

Inequality $\Big\| f_1(x) \overline{f_2(x)} |f_3 (x)|\Big\|_{H^1(\mathbb R)} \le C\|f(x)\|_{L^\infty(\mathbb R)}^2 \|f(x)\|_{H^1(\mathbb R)}$

For complex-valued functions $f_1, f_2, f_3:\mathbb R\to\mathbb C$, I want to know that the following inequality holds: $$ \Big\| f_1(x) \overline{f_2(x)} |f_3 (x)|\Big\|_{H^1(\mathbb R)} \le ...
0
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1answer
21 views

Application of the uniform boundedness principle

I have the following corollary, but I'm not sure about the final step. Assuming $Z$ to be a normed space, let $B\subseteq Z$ such that: $\forall f \in Z^*: \sup_{z\in B}\mid f(z)\mid<\infty$ then ...
6
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4answers
129 views

When is $V=U\oplus U^{\perp}$?

Let $V$ be a (infinite dimensional) vector space with inner product $(,)$ and $V$ may not be complete with the metric induced from the norm. Let $U$ be a subspace of $V$. What is the necessary and ...
6
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1answer
45 views

Which normed space have Fatou's property?

There is tool in mathematics, more specifically in Analysis, which has immense applications in handling delicates estimates, limiting arguments, etc... ; namely Fatou property. Let $E$ be a normed ...
5
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1answer
78 views

Continuous linear image of closed, bounded, and convex set of a Hilbert Space is compact

Is my proof of this proposition correct ? And is this proposition well known? Proposition: Let $C$ be a closed, bounded, and convex set in a separable Hilbert space $H$. Let $L : H \to \mathbb{R}^n$ ...
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2answers
26 views

Continuation of $W^{1,p}_0$-functions

I know that for an open set $\Omega \subset \mathbb{R}^n$ and $1 \leq p \leq \infty$, a function $u \in W^{1,p}_0(\Omega)$ can be continued to a function $v \in W^{1,p}(\mathbb{R}^n)$ by setting $v=u$ ...
3
votes
1answer
58 views

Sum of Neighborhoods of Zero

When do two neighborhoods of zero over a topological vector space add up as: $$aN+bN=(a+b)N\quad a,b\geq 0$$ I could imagine something like balanced might suffice... The problem is that I'd like to ...
1
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0answers
38 views

Absolute continuity and convolution

Suppose that $\mu$ is a finite Borel measure on the real line, $f, g\in L^1(\mu)$. Define $\nu=\mu\ast\mu$. Do I understand correctly that the convolution $f\mu\ast g\mu$ is absolutely continuous wrt ...
9
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1answer
85 views

When is an operator on $\ell_1$ the dual of an operator on $c_0$?

Suppose $T:\ell_1\to\ell_1$ is a continuous linear operator. When can we say that $T$ is a dual, or adjoint, of an operator on $c_0$? In other words, under what conditions can we find a continuous ...
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1answer
14 views

Corollary of Projection Theorem

I am having difficulty proving the following: If $M$ is a closed, proper, subspace of a Hilbert space $\mathfrak{H}$, then there exists a non-zero vector $y$ in $\mathfrak{H}$ with $y \perp M$. Any ...
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0answers
31 views

Explaining why $0,1\in S$

Please tell me if you can why $0,1\in S$, where $S:=\{\alpha\in \mathbb {C}:(\alpha x,y)=\alpha(x,y), \forall x,y\in X \}$ $(X,\Vert\cdot\Vert)$ is a generalized linear normed space; $(\cdot, \cdot): ...
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0answers
19 views

Inductive limit of topological vector space

Let $X$ be a linear space. Let a family $\{X_{\alpha}\}$ of linear subspaces $X_{\alpha}$ of $X$ be such that $X$ is the union of $X_{\alpha}$s. Suppose that each $X_{\alpha}$ is a locally convex ...
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0answers
23 views

Fixed point of projected operator

Let $X \subset \mathbb{R}^n$ be a compact convex set and let $f: X \rightarrow X$ be Lipschitz continuous and such that $$ \left( f(x) - f(y) \right)^\top \left( x-y\right) \leq 0 $$ for all $x, y ...
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0answers
28 views

Bounded below adjoint operators

Let $T\colon X\to Y$ be a bounded linear operator. Suppose that $Z$ is a subspace of $Y^*$ such that $T^*$ is [bounded below][1] on $Z$. Denote by $\text{w*-dens}\, Z$ the minimal cardinality of a ...
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26 views

About the denseness of $C^{\infty}_0 \cap W^{1,p}$ in $W^{1,p}$ [duplicate]

By the Theorem from Meyers-Serein, we know that for open $\Omega \subset \mathbb{R}^n$ and $1 \leq p < \infty$ the set $C^{\infty}(\Omega) \cap W^{1,p}(\Omega)$ is dense in the Sobolev space ...
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2answers
37 views

In a Hilbert space $H$, if the closed unit ball is compact, then how can it be proved that $H$ is finite-dimensional?

In a Hilbert space $H$, if the closed unit ball $\{x\in H\colon \|x\|\leqslant 1\}$ is compact, then how can it be proved that $H$ is finite-dimensional?
2
votes
1answer
46 views

Discontinuous seminorm on Banach space

We have known that if $X$ is a Banach space and $\sum_{n=0}^{\infty}x_n$ is an absolutely convergent series in $X$ then $\sum_{n=0}^{\infty}x_n$ is a convergent series. Moreover, we have $$ (*)\quad ...
5
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0answers
60 views

Weak*-complemented subspaces of $\ell_\infty$

Consider $\ell_\infty$ as $\ell_1^*$. Let $X$ be an infinite-dimensional complemented subspace of $\ell_\infty$ (in partiuclar, $X$ is isomorphic to $\ell_\infty$). Can we find a further subspace ...
0
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0answers
23 views

About convergence in L^1 [duplicate]

Let $(f_n)_{n \in \mathbb{N}} \subset L^1$ and $f \in L^1$ such $f_n \longrightarrow f \ a.e. $ and $ ||f_n||_1 \longrightarrow ||f||_1 $; then $ ||f_n - f||_1 \longrightarrow 0$. Why?