Tagged Questions

2
votes
3answers
77 views

Show that $c$ is closed in $l^{\infty}$

Let $$c=\{ (a_i)_{i \in \mathbb{N}} \ ; \ \ a_i \in \mathbb{R}\ ,\ \forall i \in \mathbb{N} \ , \ \mbox{exist} \ \displaystyle \lim_{i \to \infty}(a_i)\}$$ $$l^{\infty}=\{ (a_i)_{i \in \mathbb{N}} \ ...
1
vote
1answer
25 views

Good source for Triebel-Lizorkin spaces?

I'm trying to look into different types of function spaces. At the moment, at least for function spaces involving integration, I only have $L^p$ and $W^{k,p}$. The next function spaces I thought I'd ...
1
vote
1answer
33 views

How to verify whether $(C_{00},\|\cdot\|_p)$ is complete

How to verify whether $C_{00}=\{(x_n):\text{All but finitely many terms are }0\}$ is complete with respect to $p$-norm given by $$\|(x_n)\|_p=\left(\sum_{n=1}^\infty|x_n|^p\right)^{1/p}$$ where $1\le ...
3
votes
3answers
55 views

Example of two norms on same space, non-equivalent, with one dominating the other

I am looking for an example (with proof) of two norms defined on the same vector space, such that the norms on the two spaces are NOT equivalent, but such that one norm dominates the other...
4
votes
1answer
49 views

Normed vector spaces and Banach spaces

Let $X$ be a Banach space with norm $||.||$ and let $S$ be a non-empty subset in $X$. Let $F_b(S,X)$ be the vector space of $F(S,X)$ of all functions $f:S \rightarrow X$ such that $\{||f(s)||:s \in ...
3
votes
2answers
60 views

Banach spaces and quotient space

Let $X$ be a normed vector space, $M$ a closed subspace of $X$ such that $M$ and $X/M$ are Banach spaces. Any hint to prove that $X$ must be a Banach space?
-1
votes
1answer
70 views

Is $(\ell^1 , \| \cdot \| )$ a Norm space?

Suppose $ x= \{x_n \} \in \ell^ 1$ and $\| x \| = \sup | \sum_{k=1}^n x_k | $, let $ \|x\|_1 = \sum_{n=1}^{\infty} |x_n |$ is a norm for $ \ell^1 $ . Is $(\ell^ 1 , \| \cdot \| )$ a Normed ...
5
votes
1answer
80 views

Is $(l^1 ,\|.\|)$ a Banach space?

Suppose $x=\{x_n\}\in l^1$ and $\|x\|=\sup|\sum_{k=1}^{n}x_k|$, let $\|x\|_1=\sum_{n=1}^{\infty}|x_n|$ is a norm for $l^1$ . Is $(l^1 ,\|.\|)$ a Banach space?
1
vote
1answer
30 views

Proving $\ell_\infty$ is complete

I start by taking a Cauchy sequence $(a_i)$ in $\ell_\infty$. I denote the terms of $(a_i)$ as $f_1, f_2, f_3, \dots$ and so on. For each $x \in \mathbb{N}$, the sequence $(f_1(x), f_2(x), f_3(x), ...
2
votes
1answer
33 views

(p-q)-Lipschitz continuity of linear function

I have the following linear function $f(x,y,z) = ax + by + cz.$ I need to prove that f() is (p-q) Lipschitz continuous where $p=1$ and $q=\infty$. For a given two points $(x_1, y_1, z_1)$ and $(x_0, ...
1
vote
2answers
79 views

$l_1$ equipped with the sup norm is NOT a Banach Space

Prove that $l_1 = \{ x = (x_k)_{k\in\mathbb{N}}\subset \mathbb{R};\ \sum_{k\in\mathbb{N}}\ |x_k| < +\infty \}$ equipped with the norm $\| x\| = \mathrm{sup}_{k\in\mathbb{N}} |x_k|$ is NOT a Banach ...
1
vote
1answer
43 views

$L_{k}^{1}([0,1])$ is a Banach space

Let $L_{k}^{1}([0,1])$ be the space of all $f\in C^{k-1}([0,1])$ such that $f^{(k-1)}$ is absolutely continuous on $[0,1]$ (and hence $f^{(k)}$ exists a.e. and is in $L^{1}([0,1])$). Show that ...
1
vote
3answers
42 views

generalization of a normed space

I study analysis and have a problem: I have a normed space for example $(X,M)$ that is not complete, how can I complete the space $X$ with respect to norm $M$? please help me Thanks
2
votes
1answer
53 views

Banach spaces and their unit sphere

Let $X$ be a normed vector space. Show that if a subsequence of a Cauchy sequence converges, then the whole sequence converges. Use the part 1 to show that $S = \{x\in X : \|x\| = 1\}$ is complete ...
1
vote
1answer
35 views

Maximun norm over the complex sequence

Is $C_0$ (the space of all the complex sequences that satisfy $\lim_{n\rightarrow \infty }{x_n} =0$ ) is a Banach space relative to the maximum norm ( $\|x\| =max|x_n| $) and pairwise operations ? ...
4
votes
0answers
75 views

Is the result still true if we drop completeness?

I know how to prove the following exercise ( from Folland) : If $X$, $Y$ are Banach spaces. $T:X\rightarrow Y$ is a linear map such that $f\circ T\in\operatorname{dual}(X)$ whenever $f\in ...
0
votes
1answer
119 views

Isometric isomorphism

In the case that $L:B_1 \rightarrow B_2 $ is a linear mapping of Banach spaces and $L$ is a isometric isomorphism (bijection and $||Lx||_{B_1} = ||x||_{B_2} $) can I say that $L\overline{L}= 1 $ is ...
1
vote
1answer
43 views

Hypervolume of a $N$-dimensional ball in $p$-norm

Suppose I have a N-dimensional ball with radius R in p-norm: $$ \sum_{n=1}^N |x_n|^p = R^p $$ Is there a closed formula for its (hyper)volume? I can't find anything. If there isn't, can we at least ...
0
votes
1answer
173 views

About Banach Spaces And Absolute Convergence Of Seires

How to prove the following two assertions: If in a normed space $X$, absolute convergence of any series always implies convergence of that series, then $X$ is a Banach space. In a Banach space, ...
0
votes
1answer
23 views

A limit superior question in the context of the Neumann series

I'm trying to understand a step in the proof that the Neumann series converges: Let $X$ be a Banach space and $T\in L(X)$ (the space of bounded, i.e. continuous linear maps $X\to X$). It is known ...
2
votes
0answers
97 views

Is it a Banach space? If so what is its dual?

Let $(E_n)$ be a sequence of Banach spaces and $(w_n)$ be a sequence of positive real numbers. For $1\leq p <\infty$ define $\bigoplus\limits_P E_n:=\{(x_n):x_n\in E_n,\sum\limits_n\lVert ...
4
votes
2answers
127 views

Prove that $(B, \|-\|_{\infty})$ complete. B the set of bounded real valued functions on [0,1] which are pointwise limit of continuous functions.

Question: Prove that $(B, \|-\|_{\infty})$ is complete. B the set of bounded real valued functions on [0,1] which are pointwise limit of continuous functions on [0,1]. Context: Old exam problem I'm ...
1
vote
3answers
122 views

A question about the proof of the open mapping theorem in “Functional Analysis” by Rudin

In the proof of the open mapping theorem in "Functional Analysis" by Rudin, there is the following argument: Let $X$ be a topological vector space in which its topology is induced by a complete ...
2
votes
2answers
92 views

on proving that $\|\cdot\|_2$ is a norm on $C[0,1]$

Let $\mathbb{F}$ be either $\mathbb{R}$ or $\mathbb{C}$ and consider the vector space $C[0,1]$, the collection of continuous functions $f\colon[0,1]\to\mathbb{F}$. I want to show that $\|\cdot\|_2$ is ...
4
votes
1answer
225 views

Equivalence of reflexive and weakly compact

In a normed space $X$ is there an equivalence between these two proposition? 1) $X$ is reflexive; 2) $B$, the unit ball of $X$, is weakly compact.
2
votes
1answer
123 views

$C^1 [0,1]$ with different norm

If the space $C^1 [0,1]$ is equiped with norm $\Vert \cdot\Vert_1$,where $$ \Vert f\Vert_1=|f(0)|+\Vert f'\Vert _{C}=|f(0)|+\sup_{t\in [0,1]}|f'(t)| $$ for any $f\in C^1 [0,1]$, is this space Banach? ...
1
vote
1answer
39 views

Gaussian type and Euclidean sections

I have a second question about Chapter 9 in Milman and Schechtman's book "Asymptotic theory of finite dimensional normed spaces" (first question here). It's about the proof of Theorem 9.7 (pg. 55). ...
1
vote
3answers
94 views

Question about proof that multiplication in Banach algebra is continuous

Here's the proof in my notes: Where does the last inequality come from? If I want to show that it's continuous at $((x,y)$ I can use the inverse triangle inequality to get $$ (\|x^\prime\| + ...
1
vote
3answers
107 views

$B(V,W)$ is complete if $W$ is

Let $B(V,W)$ be the space of bounded linear maps from $V$ to $W$. Then it is complete with respect to the operator norm. Can you tell me if my proof is correct? Thanks. It's easy to verify that the ...
1
vote
1answer
57 views

Euclidean sections of normed spaces with known cotype

I'm having trouble digesting the proof of Theorem 9.6 in Milman and Schechtman's classic book "Asymptotic theory of finite dimensional normed spaces" (pg. 54). I'm new to functional analysis, so this ...
0
votes
0answers
131 views

$C_c(X)$ dense in $L^p$

In class we proved that $C_c(X)$ is dense in $L^p$ where $X$ is a locally compact, $\sigma$-compact Hausdorff space either equipped with a Radon measure or equipped with a locally finite measure ...
2
votes
1answer
77 views

Balls and transformed sets in normed vector spaces

Let $T$ be a surjective, continuous linear operator between two Banach spaces $E$ and $F$. Assume that it is $B_F(y_0,4c)\subset \overline{T(B_E(0,1))}$, where $c>0$, $y_0 \in F$ ($B$ is for ...
2
votes
0answers
162 views

Question about proof of Stone-Weierstrass

I would like to know if I understand the details in the proof of Stone-Weierstrass (in $\mathbb R$) so I'd like to post it here in my own words. Can you please check it and tell me if it's correct? ...
3
votes
2answers
119 views

Proof of the lemma used in proving that a finite-dimensional normed space is complete

I'm trying to understand the proof for the lemma: $$\|\alpha _1 e_1 + \alpha _2 e_2 + \cdots + \alpha_n e_n\| \geq c (|\alpha_1|+|\alpha_2|+\cdots+|\alpha_n|)$$ where $c>0$ and the $e_i$s are ...
1
vote
1answer
134 views

Question about proof of Arzelà-Ascoli

(Arzelà-Ascoli, $\Longleftarrow$) Let $K$ be a compact metric space. Let $S \subset (C(K), \|\cdot\|_\infty)$ be closed, bounded and equicontinuous. Then $S$ is compact, that is, for a sequence $f_n$ ...
2
votes
1answer
172 views

Completion of $C_c$ with respect to $\|\cdot\|_\Psi$

I'm doing the second half of the following exercise in my lecture notes: "Let $C_c(R)$ be the vector space of continuous functions $f : R \to R$ with $\mathrm{supp}(f)=\overline{ \{x \in R \mid ...
9
votes
2answers
173 views

If $V \times W$ with the product norm is complete, must $V$ and $W$ be complete?

Let $V,W$ be two normed vector spaces (over a field $K$). Then their product $V \times W$ with the norm $\|(x,y)\| = \|x\|_V + \|y\|_W$ is a normed space. Using this norm it's easy to show that if ...
1
vote
1answer
62 views

Something weaker than the Riesz basis

I have some function $f$, real valued and continuous. I formed functions $\{f_{m,k}, k \in \mathbb{Z}, m>0\}$ such that that $\mathrm{span}\{f_{m,k}, k \in \mathbb{Z}, m>0\}$ is dense in ...
2
votes
1answer
83 views

Banach space, Normed vector space

Help me please with this question. Let's $Y$ be Banach space, $Z$ - Normed vector space and $(T_{n})_{\mathbb{N}}$ - the sequence in $B(Y,Z)$ so that all sequence $(y_{n})_{\mathbb{N}}$ in Y holds: ...
3
votes
3answers
78 views

$\|\cdot\|_1 \leq \|\cdot\|_2 \leq \|\cdot\|_{\infty}$ for functions in $C([0,1])$?

Why does the following hold for continuous functions on $[0,1]$? $\|\cdot\|_1 \leq \|\cdot\|_2 \leq \|\cdot\|_{\infty}$
0
votes
0answers
45 views

existance of the interpolation space

Let $X\subset L_1+L_2$ and let $Y$ be interpolation space between $L_2$ and $L_{\infty}.$ Given $U:X\longrightarrow Y$. My question is the following: Is there exists space $Z\subset Y$, such that ...
2
votes
2answers
549 views

Prove that $X'$ is a Banach space

I'm taking a new course on functional analysis and meet with the following problem. If $X$ is a normed space (not necessarily complete), then prove that $X'$ is a Banach space. Definition: When the ...
4
votes
1answer
167 views

There exists an isometric embedding

Let $W$ be a closed linear subspace of a normed vector space $V$. Let $i_V: V \to V^{**}$. and $i_W: W \to W^{**}$ be the canonical embeddings of V and W into their second duals. Prove that there ...
2
votes
1answer
181 views

Hahn-Banach. Extend the functional by continuity

Let $E$ be a dense linear subspace of a normed vector space $X$, and let $Y$ be a Banach space. Suppose $T_{0}\in\mathcal{L}(E,Y)$ is a bounded linear operator from $E$ to $Y$. Show that $T_{0}$ can ...
37
votes
1answer
867 views

Banach Spaces - How can $B,B',B'', B''', B'''',B''''',\ldots$ behave?

(ZFC) Let $ \big\langle B,+,\cdot, \:\: \|\cdot\| \:\: \big\rangle $ be a Banach space. Define $ \mathbf{B} \; = \;\big\langle B,+,\cdot, \:\: \|\cdot\| \:\: \big\rangle $. Define $\: \mathbf{B}_0 ...
1
vote
1answer
163 views

To construct a counterexample of normed space

Please construct a counterexample for the following: $A$ is normed space and $M$ is a dense subspace of $A$, if there is a functional $f$ such that $f(M) = 0$, then $f=0$. Besides, if $A$ is a Banach ...
3
votes
1answer
691 views

Banach space of Lipschitz functions

Let $X$ be a compact metric space, and $F$ the space of all lipschitz functions $X \to \mathbf{C}$. Let $|f|_L$ be the least Lipschitz constant. We endow $F$ with the norm $||f|| = |f|_L + ...
8
votes
3answers
334 views

Is there an easy example of a vector space which can not be endowed with the structure of a Banach space

Let $V$ be a real vector space. Is there always a norm on $V$ such that $V$ is complete with respect to this norm? If not, is there an easy counterexample?
1
vote
1answer
99 views

tensorisation of linear map

Let $X$ be a Banach space and $T \colon \ell^2\rightarrow \ell^2$ be a bounded linear map. Suppose that the linear map $T\otimes Id_ {X}:\ell^2\otimes X\rightarrow \ell^2\otimes X$ which maps $e_i ...
7
votes
3answers
622 views

Operator norm on product space

I have a bilinear operator $B\colon X \times Y\to Z$ with $X,Y,Z$ normed spaces, and define a norm on $X \times Y$ by $\lVert(x,y)\rVert = \lVert x\rVert_X + \lVert y\rVert_Y$ (using the respective ...