A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.

learn more… | top users | synonyms

3
votes
0answers
45 views

Random variables that span copies of $\ell_p$

Consider the coin-toss measure $\mu$ on $\{0,1\}^\mathbb{N}$. Within this framework it is easy to construct a sequence of independent, symmetric Bernoulli random variables. Indeed the point-evaluation ...
6
votes
4answers
82 views

Definition of Equivalent Norms

Two norms $F,G$ are equivalent when there are constants $a,b$ such that $aF \le G \le bF$. I'm reading about this idea, and so far I've seen that equivalence of norms implies that the underlying ...
0
votes
3answers
35 views

Convergence on the dual of a Banach space

I have a simple question : What it means $$||v_n||_{(W^{1,p}_0)^*}\rightarrow 0$$ Where $(W^{1,p}_0)^*$ is the dual space of $W^{1,p}_0$ I know that $v_n\rightarrow 0$ in $(W^{1,p}_0)^*$ mease ...
0
votes
1answer
26 views

Dense Domain: Preimage

Given Banach spaces $X$ and $Y$. Regard a bounded operator: $$A\in\mathcal{B}(X,Y)\implies A\in\mathcal{C}(X,Y)$$ Then for dense sets: $$W\leq Y:\quad \overline{W}=Y\implies\overline{A^{-1}W}=X$$ ...
2
votes
1answer
36 views

Closed subsets of Banach space

$X=\{(a,b)\mid a \in C[0,1],b \in C[0,1]\}$, and its norm is $\|(a,b)\|=\|a\|_\infty+\|b\|_\infty.$ $Y=\{(a,a')\mid a \in C^1[0,1], \ a'(t)=\frac{da}{dt} \},\ Z=\{(0,b)\mid b \in C[0,1]\}.$ ...
1
vote
1answer
32 views

Is $C[0,1]$ equipped with $\lVert \cdot \rVert_1$ a countable union of nowhere dense sets?

Let's consider the space of all continuous function $C[0,1]$ on the intervall $[0,1]$. But instead of using the usual supremum norm we use the $L^1$-Norm $: \lVert f \rVert_1=\int_0^1 \lvert f(x) ...
0
votes
1answer
20 views

Dense Operators: Spectrum

This thread is Q&A. Given a Banach space $E$. Consider closed operators: $$T:\mathcal{D}(T)\subseteq E\to E:\quad T=\overline{T}$$ Then for the domain: ...
1
vote
2answers
29 views

Does for $T \in B(X)$ with $\|T\|>1$ exist $T^{-1}$?

Is it true if $\|T\|>1$, where $T \in B(X)$ for some Banach space $X$, then $T^{-1}$ exists? I suppose that for $\|T\|=1$ this isn't true? Because, if we suppose that inverse exists for such ...
0
votes
0answers
18 views

Showing the space of $\beta$-operators between Banach lattices is a Banach space [closed]

Let $E,F$ be a Banach lattices. A linear map $T:E \rightarrow F$ is called $\beta$-operator if $\lVert T\rVert_\beta < \infty$, where $$\lVert T\rVert_\beta := \sup \bigl\{ \bigl\Vert ...
1
vote
1answer
29 views

Please give me an example of closed subspace of banach space under some conditions

Please give me one example of Banach space $X$ and its closed subspaces $S,T,U$ which suffice following conditions. Any of $S+T,T+U,U+S$ is not a closed subspace of $X$. I can say there are some ...
0
votes
0answers
43 views

Convergence of sequence as $O(1/n)$ using damped iterations

I am trying to understand the argument for convergence here (page 16) for damped iterations of non-expansive maps. Say we have a sequence $\{x^n\}_{n=0}^{\infty}$ that is generated as $x^n = \theta ...
3
votes
0answers
63 views

When weakly compactness implies compactness?

Let $A$ be a Banach space. The weak topology on $A$ is a topology which produced by the following family of seminorms: $~~~~~~~~~~~~~~~~~~~~P_f(x)=|f(x)|,\qquad$ where $f\in A^*$ and $A^*$ is dual ...
0
votes
2answers
33 views

Fixed-point analysis similar to Banach Fixed Point Thm

I have a fixed-point question similar to the Banach fixed-point theorem. Let $f\colon \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a given function and let $x^\star \in \mathbb{R}^n$ be a known ...
0
votes
1answer
22 views

Uniform Boundedness: Nets

I thought so far that the uniform boundedness principle applies (according to the proof I know) to any net of bounded operators. Now I read that this works for sequences only. Can you shortly explain ...
0
votes
1answer
30 views

Summability: Equivalence

Summability Given a Banach space $E$. Consider sums: ...
6
votes
1answer
60 views

Do we need completeness for a weak*-convergent sequence to be bounded?

Let $(\phi_n)_n$ be a weak* convergent sequence in the dual of some normed space $X$ with (weak*-)limit $\phi$. If $X$ is Banach then it follows from the uniform boundedness principle that $\sup_n ...
1
vote
0answers
25 views

Norm of a product of projections in a Banach space

Let $X$ be a Banach space and let $P_1,P_2$ be two projections in $B(X)$, i.e., $P_1^2 = P_1, P_2^2=P_2$. My question: under what conditions do we have that $\Vert P_1 P_2 \Vert = \sqrt{\Vert P_2 ...
2
votes
0answers
29 views

Equivalent definition of uniform convexity

A Banach space $X$ is said to be uniformly convex if the following is satisfied: For $\epsilon>0, \exists \delta>0$ such that $x,y\in X, \|x\|, \|y\|\leq 1$, $\|x-y\|\geq \epsilon \Rightarrow ...
3
votes
1answer
69 views

What are some easy to understand applications of Banach Contraction Principle?

I know that Banach contraction principle guarantees a unique solution to problems of the form $$f(x) = x$$ But for the life of me I cannot understand why this problem is important at all. I don't ...
3
votes
1answer
24 views

Bessel potential space: Proof of completeness

I want to know a proof that the (one-dimensional) Bessel potential space (for $p=2$) $$H^s(\mathbb{R})=\left\{f:\mathbb{R}\to\mathbb{C}:\int_{\mathbb{R}}(1+\lvert \xi\rvert^2)^{\frac{s}{2}}\lvert ...
2
votes
2answers
78 views

Proving that a space is complete

There is something that bugs me about the proof I've been shown that $C(\Omega)$ (the space of continuos function on $\Omega$, a compact subset of $\mathbb R^n$) with the $\sup$ norm is complete. ...
3
votes
1answer
69 views

The norm of linear functional $x\mapsto \sum_{n=1}^{\infty} \frac{x_n}{2^n}$ on $c_0$

Consider the mapping $\phi :c_0 \to \mathbb{R}$ defined by $\sum_{n=1}^{\infty} \frac{x_n}{2^n}$. Compute $\|\phi\|$ Does there exist a $x \in c_0$ such that $\|x\|=1$ and $\|\phi\|=|\phi(x)|$ ...
4
votes
0answers
30 views

A question about equivalence of norms involving infimum

Let $I$ be a Banach space with norm $\lVert\cdot\rVert_I$. The norm $$\inf\{\lVert(G_i(u_i))_i\rVert_{\ell^2}\mid u=\sum_{I \geq 0}u_i\}\qquad\text{is equivalent to}\qquad \lVert{u}\rVert_{I}$$ where ...
-1
votes
2answers
36 views

Difference quotients and weak derivatives (Evans 5.8.2 theorem 3) [closed]

Could anyone give a proof on the remark below the theorem? Basically it is problem 11 I think the proof relates to example 19.19 (p.375) in enter link description here But I really do not ...
1
vote
1answer
28 views

Continuity of multiplication of operators in the strong operator topology - find an error

I need help in finding the mistake in the following reasoning. I proved that if dimension of Banach space $X$ is infinite, then multiplication of bounded operators is separately continuous but not ...
0
votes
0answers
17 views

Borel Sets and Translations

Suppose $\mathcal{F}$ is Borel $\sigma$ algebra for a separable Banach space $X$ (i.e., potentially infinite dimensional). Is it obvious that for any $A \in \mathcal{F}$ and any $a\in X$, the ...
3
votes
1answer
54 views

Differentiation calculation

$L(E)$ espace fonction continuous and linear $$\begin{array}{llll} \psi:& L(E)\times E&\longrightarrow& E\\ &(u,x)&\longrightarrow &u(x) \end{array}$$ proved the application ...
1
vote
1answer
38 views

What is the importance that an assumption needs to state whether a space is Banach space?

I am self studying functional analysis and I don't not see the utility of authors trying make it clear that a space $X$ is a Banach space before proceeding with a definition. For example, going ...
1
vote
2answers
26 views

Why does it mean for Banach space to be a locally convex topological vector spaces

A Banach space is simply a complete normed linear space. According to Wikipedia it is also a locally convex topological vector space. How does complete + normed + linear space translate into locally ...
0
votes
0answers
49 views

Prove exponential $e^f$ is of class $C^\infty$

Let $E$ a Banach space, $F=L(E,E)$ of linear and continuous functions. Define $f^0={\rm id}_E$, $f^n=f\circ\cdots\circ f$, $n$ times. Put $\exp(f)=\sum_{n=0}^\infty \frac{f^n}{n!}$. How to show the ...
1
vote
1answer
41 views

Existence of unbounded operators on Banach spaces

I'm confused by the questions Discontinuous linear functional and Example of an unbounded operator which ask about unbounded linear functionals/operators on Banach spaces. I don't understand how ...
1
vote
1answer
20 views

Passing complemented subspaces to duals

Sorry for this rather basic question from Banach space theory. Suppose I have a complemented subspace $E$ of a Banach space $X$. So let's write $i:E\to X$ to be the inclusion map. Then I have a ...
0
votes
1answer
24 views

Composition of $C^\infty$ maps between Banach spaces is $C^\infty$.

Let $V$, $W$, and $X$ be Banach spaces, and let $A \subset V$ and $B \subset W$ be open. Suppose that $F \in C^\infty(A,W)$, $G \in C^\infty(B,X)$, and $F(A) \subset B$. Is there a non-combinatorial ...
1
vote
1answer
26 views

cauchy's integral formula in operator theory

Can you prove cauchy's integral formula based on the assumptions in Conway book, operator theory, VII, 4.2? 4.2. Cauchy's Integral Formula If $\mathcal X$ is a Banach space, $G$ is an open subset ...
-2
votes
1answer
21 views

Operator not bounded below on sum of subspaces

Let $X$ be a Banach space. Say that a bounded linear operator $T\colon X\to X$ is bounded below by $\delta>0$ on $Y\subset X$ if $\|Tx\|\geqslant \delta \|x\|$ for all $x\in Y$. Is there a Banach ...
1
vote
1answer
23 views

Embedding: Extension

Problem Given Banach spaces $E_0$ and $E$ Regard dense domain: $$\overline{\mathcal{D}_0}=E_0\quad\overline{D}=E$$ Consider an embedding: ...
1
vote
1answer
36 views

How $\|a_x\| \leq c$ became $\|a_x\| \leq c\|x\|_E$?

Reading this answer here, I didn't understand the last $\color{red}{\leq}$ in: $$\Vert a(x,y)\Vert_G=\Vert a_x(y)\Vert_G\leq\Vert a_x\Vert\Vert y\Vert_F\color{red}{\leq} c\Vert x\Vert_E\Vert ...
5
votes
1answer
119 views

Every closed subspace of ${\scr C}^0[a,b]$ of continuously differentiable funcions must have finite dimension.

If $F \subset {\scr C}^1[a,b] \subset {\scr C}^0[a,b]$, then $\dim F < +\infty,$ where $F$ is a closed subspace (in $ {\scr C}^0[a,b]$). I found this answer, which is very good and solves the ...
0
votes
1answer
20 views

$p$-operator space property

If $S,T,U,V\in B(L_p(X,\mu))$, $p\in[1,\infty)$, and we regard $\begin{pmatrix} S & T \\ U & V \end{pmatrix}$ as an operator on $B(L_p\oplus_p L_p)$, then supposedly we have ...
2
votes
2answers
58 views

Banach spaces containing copies of $\ell^1$

Why is it important that a Banach spaces $X$ contains (or not) copies of the space $\ell^1$? I always hear talk about it but I don't know its importance. Could someone explain this?
-5
votes
1answer
54 views

Seminorms: Dual Space

Remarks This thread is only Q&A!* *(See guidelines: Q&A) Problem Given a Banach space $E$. Consider its dual space: $$E':=\mathcal{C}(E,\mathbb{C})\cap\mathcal{L}(E,\mathbb{C})$$ Regard ...
0
votes
1answer
35 views

Fraction of Lipschitz functions among absolutely continuous ones

Is it true that the space of Lipschitz functions on $S^1$ is a $G_\delta$ subset of the space of absolutely continuous functions on $S^1$? In which topologies ($L^p$, uniform, $C^k$, etc) it is true? ...
1
vote
0answers
21 views

A question about passing to limits in Banach lattices

I would be grateful if one could confirm that the following argumentation is fine. Suppose that $L$ is a Banach lattice and $(L_n)$ is an increasing sequence of sublattices of $L$. Given two positive ...
0
votes
1answer
36 views

Is the set defined by inequality $\|Tx\|^2\leq\|T^2x\|\|x\|$ a subspace of a Banach space?

Let $X$ be a complex Banach space and $T$ be a bounded linear operator on $X$. Put $Y=\{x\in X:\|Tx\|^2\leq\|T^2x\|\|x\|\}$. Is $Y$ a subspace of $X$? I know is that $Y$ is closed and $aY$ is ...
2
votes
1answer
59 views

Prove that the space of function $C^{k,\gamma}(\bar{\Omega})$ is a Banach space

Hi this is problem 1 in Chapter 5 Evans PDE. I have trouble showing the second part, i.e. completeness of $C^{k,\gamma}$. I know there is some previous posts but I did not quite get the answers. And I ...
0
votes
1answer
15 views

Codimension of the image of the polynomials subspace is infinite

Consider the interval $I=[0,1]$ and the Banach space $E$ of real continuous functions defined on $I$ ($E=\mathcal C_{\mathbb R}(I))$. $P \subset E$ is the subspace of polynomial functions (restricted ...
-2
votes
1answer
43 views

L(V,W) is Banach, then W is Banach [duplicate]

Let $V,W$ normed vector spaces, $V$ not empty and with a finite dimension. Prove that $L(V,W)$ is Banach, then $W$ is also Banach.
1
vote
1answer
27 views

How is completeness used in the Eberlein–Šmulian theorem?

So for weakly closed subsets of Banach spaces compactness and sequential compactness coincide, but upon studying the proof I can't put my finger on what exactly would go wrong if we dropped the ...
1
vote
1answer
29 views

Show that Quotient of a C*-algebra is a C*-algebra in its own right

Let $A$ be a commutative C*-algebra over $\mathbb{C}$ and let $J$ be an ideal of $A$ such that $J\in Closed(A)$ and that contains $x^*$ if it contains $x$, for all $x\in A$. I know that $A/J$ is also ...
3
votes
1answer
55 views

How to show that the space of polynomials is not complete

Denote by $P[0,1]$ the set of all polynomials $p\colon [0,1]\to\mathbb{R}$; this is a vector space. Endow $P[0,1]$ with the norm $$\| p\|=\sup_{t\in [0,1]}{| p(t)|}.$$ I want to show that this ...