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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412 views

Space Completion Theorem for Normed Spaces

I went to a functional analysis course this year and my lecturer wrote down this Theorem. Lots of students pointed out it is incorrect, but she insisted it was. I am stating it now and hope someone ...
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1answer
176 views

Why is the laplacian positive-definite

Let (M,g) be compact Riemannian manifold (possibly $\partial M\neq\emptyset)$ Now I have read, that "the laplace-beltrami operator is a positive definite operator". I have shown, if M is a closed ...
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1answer
92 views

A question regarding $L^1 $and $C^\infty_C(\Omega)$ spaces

$f,g \in L^1 (\Omega)$, $\Omega\subset\mathbb{R}^n$ is a Lipschitz-domain. Prove that $$(\forall\phi\in C^{\infty}_{C}(\Omega))\Big(\int_{\Omega}^{}f*\phi=\int_{\Omega}^{}g*\phi\Big)\Rightarrow f=g ...
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1answer
98 views

Solving nonlinear integral equation

Are there some known techniques for solving $h(x)=f(x)\int_0^xf(t)dt$ for $f(x)$? Are there closed form solutions?
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1answer
103 views

Inequality in Integrals

Let be $f:\mathbb{R}\rightarrow \mathbb{C}$ Why $|\int_{-\infty}^{\infty}f(x)dx|\leq\int_{-\infty}^{\infty}|f(x)|dx$? pdta:$|\cdot|$ is module of complex numbers
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3answers
458 views

Lower semi-continuous function which is unbounded on compact set.

Every lower semi-continuous functions attains an infimum/minimum on a compact set, do you know examples of lower semi-continuous functions which are unbounded and/or don't attain their ...
1
vote
1answer
97 views

$f(x)$ is a monotonic function prove the following

Prove that: $ \rightarrow\sum_{k=1}^n f(\frac{k}{n})\sum_{k=1}^n k{f(\frac{k}{n})}^2\le\sum_{k=1}^n kf(\frac{k}{n})\sum_{k=1}^n{f(\frac{k}{n})}^2 $ Given $f(x)$ is a positive function and also ...
3
votes
1answer
116 views

Positive operator is bounded

For a real Banach space $X$ let $A:X\rightarrow X^*$ be a positive operator in the sense that $(Ax)(x)\geq 0$ for all $x\in X$. Show that $A$ is bounded. I don't know how to do that, maybe it's ...
2
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1answer
198 views

Positivity of the anti-commutator of two positive operators implies commutativity?

This is a generalization of question Positivity of the anti-commutator of two positive operators . note: by positive operator, I mean positive semidefinite (i.e. $\ge 0$, not necessary $>0$). Let ...
1
vote
1answer
320 views

Equality in the Cauchy-Bunyakowsky-Schwarz inequality for a semi-inner product

I am stuck with an exercise that I found in a textbook by Conway. First, I would like to clarify what is meant by a semi-inner product. Definition. Suppose that $\mathscr X$ is a vector space over ...
2
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0answers
139 views

Sobolev-type inequality.

Let $0<\alpha<n$, $1<p<q<\infty$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$. Then: $$\left\|\int_{\mathbb{R}^n} \frac{f(y)dy}{|x-y|^{n-\alpha}} \right\|_{L^q(\mathbb{R}^n)} \leq ...
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0answers
423 views

Confused by a proof in Rudin *Functional Analysis*

I am reading Rudin's Functional Analysis and got quite confused by his proof of Thm 8.5, that is, the existence of fundamental solutions for differential operator $P(D)$, where $P$ is a polynomial. ...
0
votes
1answer
45 views

Alternative proof for not attained norm

Consider a mapping $$T_\lambda: \ell^1 \rightarrow \ell^1\quad T_\lambda f:=\{\lambda_1 f_1,\,\lambda_2 f_2,\lambda_3 f_3,\,\cdots\},$$ where $\lambda_n = 1 - \frac{1}{n}$, $\lambda \in \ell^\infty$. ...
2
votes
2answers
87 views

characterisation of compactness in the space of all convergent sequences

I go through a proof of the following. Let $(\ell_1,d)$ be the metric space of all sequences $x = (\xi_i)_{i \in \mathbb{N}}$ with $\sum_{i=1}^{\infty} |\xi_i| < \infty$ and the metric $$ d(x,y) ...
4
votes
2answers
924 views

$\ell_{p}$ space is not Hilbert for any norm if $p\neq 2$

My question is motivated by this one: $\ell_p$ is Hilbert space if and only if $p=2$ Maybe it is a simple thing or im just confused but, suppose we are given any norm in $\ell_{p}$ for $p\neq 2$. ...
2
votes
1answer
1k views

$\ell_p$ is Hilbert space if and only if $p=2$

Can anybody please help me to prove this.. Let p greater than or equal to 1,show that the space of all p-summable sequences is an inner product space if and only if p=2
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0answers
89 views

Bound from Frechet derivative

I am not certain about a step in the following argument where I use the frechet derivative. Problem Let $F$ be a continuously differentiable function with founded derivative (let $f$ be its ...
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1answer
150 views

functional analysis-inner product space of $C[a,b]$

Prove that $C[a,b]$ is a inner product space.. Please help me to prove the following axioms or can anybody send me a link which includes these proofs. 1.conjugate symmetry property 2.inner product ...
0
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1answer
189 views

Two equivalent characterisations of totally bounded (pre-compact) sets

Let $A$ be a subset of a metric space $(X,d)$, I want to show equivalence of (i) For every $\varepsilon > 0$ there is a finite set $E = \{ x_1, x_2, \ldots, x_n \} \subset X$ with $$ A \subset ...
0
votes
1answer
123 views

Characterizing positive semi-definite operators in $\mathcal{B}(L^2)$

I am asking perhaps a stupid question. How can I characterize all positive semi-definite operators in $\mathcal{B}(L^2(X,\lambda))$, where $\lambda$ is the Lebesgue measure. For a start, let us ...
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votes
2answers
57 views

Quick criterion to decide whether a limit of functions in $W^{1,p}(\Omega)$ is in that space

Let $\Omega\subset \mathbb R^N$ be an open set. Suppose we are given a sequence $u_n$ in $W^{1,p}(\Omega)$, $1<p\leq+\infty$, such that $u_n\to u$ in $L^p(\Omega)$ and such that $(\nabla u_n)$ is ...
0
votes
1answer
121 views

Norm piecewise continuous function

Let be $f:\mathbb{R} \rightarrow \mathbb{C}$. Consider the space of piecewise linear curves, with support in the interval [-1,1], sucht that $f(x)= A-|x|$ if $|x|\leq 1$; $f(x)= 0$ otherwise. For this ...
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1answer
107 views

How to show Wiener measure induces basic properties of Brownian motion?

page 19 of http://www.math.tifr.res.in/~publ/ln/tifr64.pdf gives a defintion of Wiener measure Ft1,t2,..,tk. But how can we show it is a probability measure and it satisfies the consistency condition ...
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1answer
181 views

Figure out wheter or not these transformations are linear and bounded

In order to check wheter a transformation is linear and bounded i need to show that $$T(ax +by) = aT(x) + bT(y)$$ But when the examples get harder, I have trouble doing so. For instance the ...
2
votes
0answers
131 views

compactness of Banach spaces

I'm stuck on the following problem: a) Let X be a Banach space, $K \subset X$ non-empty and compact and $\epsilon > 0$. Show that $K^{\epsilon} := \{k+x | k \in K, \|x\| \leq \epsilon\}$ is not ...
8
votes
1answer
670 views

Reconciling several different definitions of Radon measures

Upon reviewing some basic real analysis I have encountered two different definitions for Radon measure. Let the underlying space $X$ be locally compact and Hausdorff. Folland's Real Analysis gives the ...
3
votes
1answer
83 views

Pseudo Monotone Operator

Suppose $X$ is a real Reflexive Banach space. Let $A:X\rightarrow X^{\star}$ be a Pseudo Monotone operator, i.e. if $u_{n}\rightharpoonup u$ and $\limsup\langle Au_{n},u_{n}-u\rangle\leq 0$, then ...
0
votes
0answers
50 views

functional analysis question: extended version of completion of $L_2$ space

In $L_2$ space, only use domain [0, 1]. Prove, for any $C^2$ function F in $L_2$, there exits a sequence of polynomials ($f_k$) such that $f^{(r)}_k$ tends to $F^{(r)}$ for r=0,1,2 uniformly. Note: ...
4
votes
1answer
179 views

1-separated sequences of unit vectors in Banach spaces

Given an infinite-dimensional Banach space $X$, I would like to construct a sequence of linearly independent unit vectors such that $\|u_k-u_l\|\geqslant 1$ whenever $k\neq l$. Any ideas on how to ...
3
votes
1answer
118 views

Infimum of the spectrum of an unbounded selfadjoint operator

Let $A$ be an unbounded selfadjouint operator in the Hilbert space $H$, having domain $D(A)$. Denoting by $\sigma_A$ the spectrum of $A$, we have $\inf \sigma_A \ = \ \inf_{u\in D(A),\|u\|=1} \ ...
0
votes
1answer
67 views

$ f \in W^{s,2}$ then $ \int_{\Bbb R^n} \xi_j^{2s} | \mathscr F f( \xi) |^2 d \xi < \infty $?

If $f \in W^{s,2} (\Bbb R^n) $, then by the Plancherel's theorem, I know that its Fourier transform $ \mathscr F f(\xi) \in L^2 (\Bbb R^n) $. ($ \scr F$ means the Fourier transform). Now I want to ...
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vote
1answer
205 views

Topological vector space generated by weak topology

Q1. Let $(X, \|.\|)$ be a real Banach space and $\tau$ is the weak topology on $X$. I would like to ask whether $(X,\tau)$ is a topological vector space? Q2. Let $(X, \|.\|)$ be a real Banach space ...
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vote
2answers
488 views

About the sum of two sets in a topological vector space

Let $X$ be a TVS and $A\subseteq X$. Then it is known that for any open set $B$ in $X$, the set $A+B$ is also open. In particular, the sum of two open sets is again open. In Rudin's book in Functional ...
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votes
2answers
206 views

show that $(-1,1)$ and $\mathbb{R}$ are isometric

I want to find a metric $d'$ sucht that $(\mathbb{R}, d')$ and $(X, d)$ with $X = (-1,1)$ and $d(x,y) = |x-y|$ are isometric. I tried the homeomorphisms $$ f:\mathbb{R} \to (-1,1) ~ \textrm{ with } ~ ...
0
votes
1answer
78 views

Discrete sets in normed spaces

Let $D$ be a discrete set in the norm topology of a normed space $X$. Does $D$ remain discrete in the weak topology of $X$? What if $D$ is moreover closed?
2
votes
2answers
191 views

Separation in dual space

Let $X$ be a real Banach space and $X^*$ its dual space. Let $C^*$ be a weak$^*$ closed and convex subset in $X^*$ and $x^*\notin C^*$. Then there exists $x\in X$ such that $$ \langle x^*, x\rangle ...
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vote
1answer
149 views

Sum of weak star closed set and compact weak star set

Let $X$ be a real Banach space and $X^*$ be its dual space. Let $C$ be a weak$^*$ closed subset in $X^*$ and $D$ a compact weak$^*$ in $X^*$. I would like to ask whether $C+D$ is closed weak$^*$ in ...
0
votes
1answer
69 views

exist $a \in \mathbb{R^n}$ such that $f(a;v)=0$

Seja $f: \mathbb{R}^n \to \mathbb{R}$ a continuous function that have all directional derivates in any point of $\mathbb{R^n}$. If $f'(u;u)>0$ exist for all $u \in S^{n-1}$, prove that exist $a \in ...
2
votes
1answer
100 views

About the continuity of the Fourier transform.

If $ x^{\alpha} g \in L^1 ( \Bbb R^n)$ for $| \alpha | \leqslant k$, then how can I prove that its Fourier transform $$ \mathscr{F} g \in C^k ( \Bbb R^n) ?$$ Here $\alpha$ is a multi-index.
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votes
2answers
483 views

Orthonormal basis for Sobolev Spaces

Sobolev spaces of order 2 are known to form a Hilbert space. Consider such a Sobolev space of (order 2) functions on the domain $f:\mathbb{R}\rightarrow \mathbb{R}$. What is an example for the basis ...
2
votes
1answer
64 views

If $ f \in L^2 ( \Bbb R^n)$ , then $\lim_{x_k \to \infty} f(x_1 , \cdots , x_k , \cdots , x_n ) = 0$ ?

If $ f \in L^2 ( \Bbb R^n)$ , then $\lim_{x_k \to \infty} f(x_1 , \cdots , x_k , \cdots , x_n ) = 0$ ? If this is true, then how can I prove this?
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2answers
464 views

strange metric $d(x,y) = ||x|| + ||y||$ if $x\ne y$, $d(x,y) = 0$ if $x = y$.

Let $d : \mathbb{R}^n \times \mathbb{R}^n \to [0, \infty]$ be defined by $$ d(x,y) = \left\{ \begin{array}{ll} 0 & : ~ x = y \\ ||x|| + ||y|| & : ~ x \ne y \end{array} \right. $$ where ...
4
votes
0answers
160 views

Haar Measure: Unimodular Locally Compact Groups

I have the following problem: "Let $G$ be a locally compact group, all of whose normal subgroups are contained in $Z(G)$. Prove that $G$ is unimodular." My attempt at attacking the problem was to ...
5
votes
1answer
858 views

Norm of integral operator in $L^1$

What is the norm of the operator $$ T\colon L^1[0,1] \to L^1[0,1]: f\mapsto \left(t\mapsto \int_0^t f(s)ds\right) $$ ?
2
votes
1answer
395 views

How to prove that the sum of two compact sets in a Banach space need not be compact

Let $X$ be a Banach space and $K$ a compact subset of $X$ and consider for a given $\eta>0$ the closed ball $C(0,\eta)$ centered at $0$ of radius $\eta$. How can I show that $K+C(0,\eta)=\{x+y: ...
1
vote
1answer
187 views

weak convergence condition

Let $l^{2}=\left\{x=(x^{(1)},x^{(2)},...):\sum_{i=1}^{\infty }\left\vert x ^{(i)}\right\vert ^{2}<\infty \right\} $. Would you help me to prove that $({\vert|x_n |\vert})$ is bounded sequence and ...
3
votes
1answer
152 views

Closure of $l_1$ in $l_\infty$

Suppose we have a set $A$ which is the set of all sequences that satisfy $|x_n|\xrightarrow{} 0$. If we consider $l_1$ to be a subset of $l_\infty$. Show that the closure of $l_1$ in $l_\infty$ equals ...
2
votes
2answers
408 views

Norm of a linear functional

Consider $C[0,1]$ (the space of continuous functions on $[0,1]$) with the max-norm (assume the underlying field is $\mathbb{R}$). For $g \in C[0,1]$, define $\Phi_g: C[0,1] \rightarrow \mathbb{R}$ by ...
1
vote
0answers
71 views

Inductive limits

Let $E_n$ be a family of Banach spaces. Under which conditions imposed on $(E_n)$ can we represent the $\ell_\infty$-sum $(\bigoplus_{n\in \mathbb{N}} E_n)_{\ell_\infty}$ as a complemented subspace of ...
1
vote
1answer
119 views

Frames for Hilbert space

A sequence $\{f_{n}\}_{n\in I}$ is a frame for a separable Hilbert space $H$ if there exists $0<A\leq B<\infty$ such that $$ A\|f\|^{2} \leq \sum_{n\in I}|\langle f,f_{n}\rangle|^{2}\leq ...