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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107 views

This is Banach space?

Let me denote by $C_{0}(\mathbb{R})$ the set of continuous functions which tend to zero at + and - $\infty$. I am wondering if it is true that $(C_{0}(\mathbb{R}), \Vert . \Vert_{\infty})$ is a ...
6
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0answers
63 views

Radon-Nikodým (write the density as a limit)

Let $\mu$ be a probability measure and $\nu$ a $\sigma$-finite measure on $(\mathbb{R},\mathcal{B})$ with $\nu\ll\mu$. Show that it is $\mu$-a.s. $$ \lim_{h\to 0}\frac{\nu [x-h,x+h)}{\mu ...
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1answer
31 views

Example for a two norms and vector space which are equivalent on this Vectorspace

I need an example where the two norms and the vector space are equivalent on the vector space.
2
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1answer
147 views

Structure of the functional space $\int_ {- \infty} ^ \infty f (x) dx = 1 $

Please, help me with studying of useful practical features of the following functional space: $$\int_{-\infty}^\infty f(x) \, dx = 1$$ For example: 1) What basis types are most convenient for ...
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0answers
21 views

Proving $\overline{f}(x)\leq g(x)$ in Hahn-Banach Theorem

I'm trying to prove the Hahn-Banach Theorem. We have $Z\subset X$. The sublinear functional on $X$ is $p$, and the linear functional on $Z$ is $f$. In the proof, I have already found the maximal ...
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1answer
80 views

proving existence of diffeomorphism

In my hand out of manifold, I found the following lemma but there is no proof there: Let $U\subseteq\mathbb{R}^m$ be open and pick some $a\in U$. Suppose that $f:U\mapsto \mathbb{R}^n$ is a smooth ...
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1answer
150 views

$c_0$ is a closed subspace of $l^{\infty}$

Put $$ l^{\infty} = \{ (x_n) \subseteq \mathbb{C} : \forall j \; \;\ \;|x_j| \leq C(x)\} $$ I want to show that $c_0$, the space of all sequences of scalars that converges to $0$ is closed subspace ...
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1answer
94 views

Support of a function and the struggle…

The concept of support is very confusing to me, I'm just getting used to it. Lets consider $\Omega\subseteq\Bbb R^n$ an open set, $C^1_c(\Omega):=\{f\in C^1(\Omega)\mid\operatorname{supp}(f)\text{ is ...
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1answer
53 views

Situation of Nirenberg-Sobolev embedding

Suppose $f\in L^2(\mathbb{R}^2)$ with compact support and $\frac{\partial ^{(k)}} {\partial x^{K}}f, \frac{\partial ^{(k)}} {\partial y^{K}}f\in L^2(\mathbb{R}) \; \forall k\in\mathbb{N}$. Can we ...
3
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1answer
199 views

Use of cut-off functions and partitions of unity

This is a simple problem, but I would still be very thankful if you could give me an advice on it. I'm trying to show that in a compact M-dimensional manifold, $$\int e^w \sqrt{g}\, dx \leq C \exp ...
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2answers
43 views

Show this set forms an ellipse in R2

I'm trying to show that $\{ (x, y) : \|x\vec v +y \vec w \| = 1 \}$ in $\mathbb{R}^2$, with $\vec v,\vec w$ elements of a real inner product space, is the equation of an ellipse centered at $\vec 0$. ...
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3answers
67 views

Norm of vectors inequality

I tried proving this with triangular inequality but i was not right can any one help me with this
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1answer
54 views

$l^p$ is super subset of $ l^q$ if and only if?

$$ l^p \subseteq l^q. $$ if and only if ? $l^p$ is a subset of $l^q$ when is it possible ? !
2
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1answer
45 views

Properties of a linear operator given an inequality

Let $X$ be a Banach** space and let $X'$ be its dual. Let $x_0 \in X$ and assume that there is $L \in X'$ such that for every $x \in X$ $$\frac{1}{2}\|x_0\|_X^2 - L(x_0) \le \frac{1}{2}\|x\|_X^2 - ...
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2answers
79 views

How to prove $l^3(\mathbb R)$ is a vector space.

I have a small confusion,i know how to prove $l^2(\mathbb R)$ is a vector space but i am not getting any idea to prove $l^3(\mathbb R)$ is a vector space. Vector space is defined as
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2answers
123 views

The closure of $C^{1}_{0}(\mathbb R)$ in $L^{\infty}$?

How could we see that the closure of $C^{1}_{0}(\mathbb R)$ in $L^{\infty}$ is the set of continuous functions that go to zero at + and - $\infty$. It is absolutely not clear for me why it is this ...
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0answers
196 views

Convolution of piecewise function

I would like to compute the convolution of piece wise function Following is the piecewise function $$ C_a(t) = \begin{cases}0& t\leq t_d\\ ...
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2answers
155 views

Density of $C^{1}_{0}(\mathbb R)$ in $L^{\infty}(\mathbb R)$

I am looking for a counterexample to $C^{1}_{0}(\mathbb R)$ ( $C^1$ functions with compact support) is dense in $L^{\infty}(\mathbb R)$? Is there some easy counterexample showing that this latter is ...
2
votes
1answer
41 views

Norm of vector with respect to operator

Define $L$ is a linear operator maps from $E^n$ to $\mathbb{R}$, its norm is defined as $||L||_{op}=\sup\limits_{||x||=1}L(x)$, where $||\cdot||$ is any norm on $E^n$. How to show that ...
2
votes
2answers
156 views

A closed ideal in a commutative Banach algebra $C(X)$

Suppose that $A$ is a natural Banach function algebra on $K$, a compact Hausdorff space. So $A$ is realised as an algebra of continuous functions on $K$, is a Banach algebra for some norm (necessarily ...
2
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0answers
65 views

Dual Lorentz characterization of $L^q$

I am trying to do Exercise A.4 of Terence Tao's book Nonlinear dispersive equations: local and global analysis. The exercise is on p.342 in Appendix A. The problem statement is: Let $f\in ...
6
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0answers
166 views

renorm a Banach space to make an operator have spectral radius equal to norm

Let $X$ be an infinite-dimensional complex Banach space equipped with the norm $\lVert\cdot\rVert$, and let $T\in\mathcal{L}(X)$ a bounded linear operator on $X$. Let $r(T)$ denote the spectral ...
2
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0answers
39 views

The space of bounded mean oscillation $BMO(B_R)$, live in the Campanato space $\mathcal{L}^{1, n}(B_R)$

Let $B_R$ be an open bounded ball in $\mathbb{R}^n$. I am trying to show that if $u\in BMO(B_R)$ then $u\in \mathcal{L}^{1, n}(B_R)$ and that \begin{equation} \|u\|_{\mathcal{L}^{1, n}(B_R)}\leq ...
2
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1answer
206 views

Linearly independent functionals

Let $ f_1,\ldots,f_n$ be linearly independent linear functionals on a vector space $X$. Show that there are $n$ elements $x_1,\ldots,x_n$ in $X$ such that the $n\times n$ matrix $[f_i(x_j) ]$ is ...
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1answer
76 views

A question about Hahn-Banach theorem.

Hahn-Banach theorem states: Let $X$ be a real vector space and $p$ a sublinear functional on it. Also, let $Z\subset X$ be a subspace, and $f$ a sublinear functional on $Z$. Also, for all $z\in ...
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1answer
125 views

Is the unit sphere in $(C[0,1], \| \cdot\|_1)$ compact?

Consider the normed space $(C[0,1], \| \cdot\|_1)$ where $C[0,1]=\{f:[0,1] \to \Bbb R : f$ is continuous$\}$ and $\|f\|_1 = \int_0^1|f(t)|dt$. I'm trying to find out if the unit sphere $S=\{f \in ...
3
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2answers
163 views

Why is the set of all $\infty$-tuples with finitely many non-zero rational terms dense in $\ell_2$?

This statement has been given as an example in the book "Introductory real analysis" written by Kolmogorov and Fomin: The set of all points $x=(x_1,x_2,\cdots,x_n,\cdots)$ with only finitely ...
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0answers
42 views

Linearly independent linear functionals [duplicate]

Let $ f_1,\ldots,f_n$ be linearly independent linear functionals on a vector space $X$. Show that there are $n$ elements $x_1,\ldots,x_n$ in $X$ such that the $n\times n$ matrix $[f_i(x_j) ]$ is ...
2
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3answers
137 views

The dual space of $c$ is $\ell^1$

Here is what I know/proved so far: Let $c_0\subset\ell^\infty$ be the collection of all sequences that converge to zero. Prove that the dual space $c_0^*=\ell^1$. $Proof$: Let $x\in c_0$ and let ...
3
votes
1answer
250 views

If $f \in L^{\infty}$ and $\exists r < \infty$ so that $\|f\|_r < \infty$, show $\lim_{p \rightarrow \infty} \|f\|_p = \|f\|_{\infty}$ [duplicate]

Question: This is the last part of a 5 part question I am working on. Let $(X,\mu)$ be a possibly infinite measure space. Assume $\exists r < \infty$ with $\|f\|_r < \infty$ and that $\|f ...
4
votes
1answer
219 views

Prove that the weak$^*$ topology on the space of tempered distributions is not 1st countable

Please, help me with a proof of this (apparently) known fact whose proof is out of my reach, even though I spent a considerable amount of time looking it up: The weak$^*$ topology on the space of ...
0
votes
1answer
39 views

Is this a valid argument to show $|a|^q \in L^{(p/q)'}(\Omega)$?

Brezis' functional analysis book includes a solution to one of its exercises that is looking somewhat fishy to me: while attempting to formalize the last step in the solution, I have arrived at a ...
3
votes
3answers
194 views

Necessity of completeness of the inner product space in Riesz representation theorem

I wanted to find a counter example to show that the completeness of the inner product space is necessary in Riesz representation theorem. Please give an example of a bounded linear functional $T$ on ...
3
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0answers
63 views

$f, f|f| \in L^{1} (\mathbb R) \cap C_{0} (\mathbb R) \implies |f| \in H_{1} (\mathbb R)$?

Suppose $f \ \text{and} \ f|f|\in L^{1}(\mathbb R).$ Then, clearly, $|f|\in L^{2}(\mathbb R)$ and therefore by Plancheral theorem, we get, $\widehat{|f|} \in L^{2}(\mathbb R).$ Also, assume, $f, ...
2
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0answers
66 views

How to show that the canonical double-dual map $c_0\to\ell^\infty$ is not surjective?

I have a question regarding dual spaces. We know that the dual space of $\ell^1$ is isomorphic to $\ell^\infty$, and the dual space of $c_0$ is isomorphic to $\ell^1$. Here $c_0$ refers to the normed ...
0
votes
1answer
45 views

Subspaces of finite dimensional Hilbert spaces

This might be a trivial question but please point out exactly where my reasoning is incorrect. Is every subspace of $\mathbb{R}^n$ closed since $\mathbb{R}^n$ with the dot product is a finite ...
2
votes
1answer
122 views

Extremum of functional of a complex function

consider functional $E$ defined by $$E[z]=\int F(x,z(x))dx$$ where $F$ is a complex-valued nonlinear function. How can we find the function $z(x)$ so that $$G=|E|^2=EE^*=\iint ...
1
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2answers
165 views

Weak convergence of norms of sequence

Let $x_n \to x$ weakly. My question is: does it hold that $\|x_n\|\to \|x\|$? I haven't been able to work out the answer and I'd appreciate help with it but here are my thoughts: Given the inverse ...
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1answer
52 views

Definition of $L^p$ as a set

Looking at Functional Analysis, I know that the space of convergent sequences $\ell^p$ is defined to be $$\ell^p := \left\{\left(x_k\right)_{k=1}^\infty\mid\sum_{k=1}^\infty ...
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1answer
240 views

Homeomorphisms between infinite-dimensional Banach spaces and their spheres

As I know Cz. Bessaga has proved that an infinite-dimensional Banach space is homeomorphic to its unit sphere. Unfortunately I do not have his book but I want to know is this theorem true without ...
0
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1answer
41 views

WOT Continuous operators

Let $E$ be a Banach space, I'm trying to figure out the trick to prove that all the WOT-continuous operators $\phi:B(E)\to\mathbb{C}$ are of the form $$\phi(T) = \sum_{i=1}^{n}\varphi_{i}(Tx_{i})$$ ...
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1answer
61 views

On the isomorphism between bounded sesquilinear forms and bounded operators between two Hilbert spaces

Let $H$ and $K$ be two Hilbert spaces. Let $S(H,K)$ be the vector space of bounded sesquilinear forms $u:H\otimes \overline{K}\to\mathbb{C}$, and let $B(H,K)$ be bounded linear operators from $H$ to ...
3
votes
1answer
99 views

Are these two steps in this proof necessary?

Theorem: Let $A$ be a unital Banach algebra. Then for $a \in A$ the spectrum $\sigma (a) \neq \varnothing$. Consider the following proof: The first step that seems unnecessary to me: Let's say we ...
0
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0answers
108 views

Convergence problem in Banach spaces

We know that if $X$ is a normed linear spaces, the strong convergence of a sequence implies the weak convergence. Suppose that $X$ is a normed linear spaces. An element in $X$ is denoted by $x$ and ...
1
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1answer
70 views

Approximation of bounded Borel functions

Let $K$ be a compact space and let $B$ be the space of bounded Borel functions on $K$ equipped with the supremum norm. Show that simple functions (i.e. functions attaining only a finite number of ...
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3answers
45 views

Notation of this set in a set?

I am currently struggeling with the following notation: For $\epsilon \in (0,1)$ and $p \in (0,\infty)$, consider the following subset of $L ^p$: $M(p,\epsilon)=\{f \in L^p:m \{x:|f(x)| \ge \epsilon ...
0
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1answer
97 views

infinite dimensional normed space is not complete

I want to show that space X with norm of sum is not complete. for any x, we have $x= \sum^n_{k=1}c_{k}b_k$ where $b_k\in basis$ and $c_k\in field$ norm is $\lvert\lvert x \rvert\rvert = $ ...
2
votes
1answer
117 views

Behavior of the resolvent near the boundary of the spectrum

My question is, in some sense, a continuation of the question below. Isolated singularities of the resolvent Suppose $T\in B(H)$ has no eigenvalues, pick $x\in H$, $x\neq 0$, and consider the ...
0
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1answer
35 views

How to find the quotient space $C^p[0,1]$/$C^p_{F}[0,1]$

How to find the quotient space $C^p[0,1]$/$C^p_{F}[0,1]$ $X=C^p[0,1]=\lbrace x : \exists x^{(p)}\in C[0,1] \rbrace$ $C^p_{F}[0,1]=\lbrace x\in X : x^{(k)}(0)=0 $ for $ 0\le k\le p \rbrace$
0
votes
1answer
41 views

How to find the quotient space of $ C[0,1]$/$C_{0}[0,1]$

How to find the quotient space of $ C[0,1]$/$C_{0}[0,1]$ $C_{0}[0,1]= \lbrace x\in X : x(0)=0 \rbrace$ it will look like that $\forall a\in \mathbb{R}$ , $\lbrace x\in C[0,1] : x(0)=a \rbrace $. so ...