Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [banach-spaces]

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

1
vote
1answer
32 views

A question on the dual of $l_\infty(X)$

Let $X$ be a compact Hausdorff space. Let $C(X)$ be the continuous functions on $X$ with sup norm and $l_\infty(X)$ be the bounded functions on $X$, also with the sup norm. Suppose $\{g_n\}_{n\in\...
2
votes
2answers
25 views

Showing that $(Y, \| \cdot \|)$ is Banach, iff $Y \subset (X,\| \cdot \|)$ is closed.

Exercise : Let $Y$ be a subspace of the Banach space $(X, \| \cdot \|)$. Show that $(Y, \| \cdot \|)$ is Banach iff $Y$ is closed. Question : Any tips or hints on how to start this ? I see myself ...
2
votes
0answers
36 views

Confusion between dual of continuous functions and borel functions

Let $X$ be a compact Hausdorff space and let $$C(X) = \{\text{ Continuous complex functions on } X\}$$ $$B(X) = \{ \text{Bounded complex Borel-measurable functions on X}\}$$ both equipped with the ...
-3
votes
0answers
18 views

Showing that $L_p[a, b]$ is a separable Banach space [on hold]

Prove that For each $p\ge1, L_p[a, b]$ $($here $L_p[a, b] =l_p[a, b]/E_o = { [f] : f \in l_p [a, b] }, E_o\subset E$, where $E$ - normed linear space and $l_p [a, b] = {f : [a, b] \to R, \...
1
vote
0answers
17 views

About the equivalence of two definitions of topological linking

If $E$ is a Banach space, denote with $I$ the identity map of $E$, denote the space of continuous functions from $E$ to $E$ with $\operatorname{C}(E,E)$ and denote the space of homeomorphism from $E$ ...
3
votes
1answer
31 views

Show that if the norms $\| \cdot \|_1$ and $\| \cdot \|_2$ are equivalent, then $(X, \| \cdot \|_1)$ is Banach iff $(X, \| \cdot \|_2)$ is Banach

Exercise : Show that if the norms $\| \cdot \|_1$ and $\| \cdot \|_2$ are equivalent, then the space $(X, \| \cdot \|_1)$ is a Banach space if and only if the space $(X, \| \cdot \|_2)$ is a Banach ...
0
votes
0answers
20 views

Can the existence of a Schauder basis help in the verification of the convergence of a sequence of operators?

I was wondering if the existence of a Schauder basis helps in the verification of the convergence of a sequence of operators. The following is straightforward: Let $(T_n), T\in \mathcal{L}(X)$, ...
2
votes
1answer
35 views

Are Baire class functions closed under pointwise limits?

I am confused about the notion of Baire functions (real or complex valued) on a compact space $X$. The set of Borel functions on $X$, $Bo(X)$ is defined to be the set of those functions $f$ for ...
1
vote
2answers
39 views

Continuous map but not bounded on Banach space

Let $X$ be a Banach space. I am looking for an example of a function $f\in C(X)$ but $f\notin B(X)$, i.e., there exist a bounded set $S\subset X$ such that $f(S)$ is not bounded in $X$. Note that ...
0
votes
2answers
27 views

Showing compact operator

Let $X,Y,Z$ be Banach spaces and let $T:X\to Y$ be a compact linear operator, $S:Z\to Y$ be a bounded linear operator such that $S(Z)\subset T(X)$. I have to show that $S$ is a compact linear operator....
1
vote
0answers
34 views

Does Grothendieck's theorem hold for Bounded borel functions?

In the case of continuous functions on a compact Hausdorff space, we have that any bounded set is pointwise compact if and only if it is weakly compact and the two topologies coincide with this set. ...
1
vote
1answer
32 views

Compute the norm of a bounded linear operator

Let $T$ be a nonzero bounded linear operator in $B(H)$, where $H$ is an infinite dimensional Hilbert space. If the norm of $T$ is known, how to compute the norm $\|I–T\|$,where $I$ is the identity ...
0
votes
0answers
14 views

Continuous and diferentiable map going onto Banach space

I'm trying to prove the following lemma but I can't finish it: Let $f:[0,1] \rightarrow F$ continuous and let $F$ be a banach space. If $f$ is diferentiable on $(0,1)$ and $|| f'(t)|| \leq M$ for ...
0
votes
2answers
21 views

The relation of type $A \subset V \subset \overline{V} \subset U$ in Banach space

Let $X$ be a Banach space, $A$ is a compact subset of $X$, $U$ is an open set containing $A$. Question: Does there exist a relatively compact open set $V$, such that $A \subset V \subset \overline{...
1
vote
0answers
11 views

Banach spaces partial derivatives [Proof verification]

I want to prove the following proposition but I'm not sure my proof is correct. I would apreciate if someone can check if it's correct, thanks. Let $U \subset E=E_1 \times E_2 \dots\times E_n$, ...
3
votes
1answer
55 views

Is every locally Banach, Hausdorff space regular?

I am working on some infinite dimensional differential geometry. I have tried proving a somewhat weaker statement than the above by replacing locally Banach with locally metrizable. But after some ...
2
votes
0answers
18 views

Matrix of compact operator on Banach space of the form $\ell^p(Z,\ell^p)$

For a countable subset $Z$ of a metric space $X$, consider the Banach space $\ell^p(Z,\ell^p)$ for $p\in[1,\infty)$. The space of continuous functions $C_0(X)$ acts on $\ell^p(Z,\ell^p)$ by ...
0
votes
1answer
51 views

Open sets in infinite dimensional spaces

Let $C$ be an closed subset of the Banach space $X$. I am wondering whether the following statements are equivalent for any Banach space: $O$ is an open set containing $C$. For every $x\in \partial C$...
2
votes
2answers
29 views

Norm induced by Banach space and a Linear Transformation

Let E be a normed vectorial space on $\mathbb{K}$ and F a vectorial space on $\mathbb{K}$ . $L : E \rightarrow F$ a surjective linear operator such that $L^{-1}(0)$ is closed. Prove that $$\|y\| = \...
1
vote
1answer
31 views

Can I use Banach-Steinhaus?

Let $X$ be a Banach space and $(x_n)_n$ a subset of $X$ such that$$\sum_{j\geq1}|\langle\phi,x_j\rangle|<∞.\quad\forall\phi\in X^*$$Show that$$\sup_{\|\phi\|=1}\sum_{j\geq1}|\langle\phi,x_j\rangle|&...
1
vote
1answer
17 views

Compactness of operators in Banach spaces

Let $X,Y,Z$ be Banach spaces. Let $T \in \mathbb L (X, Y)$ be a compact operator. Why are the operators $B_1 \circ T \in \mathbb L(X,Z)$ and $T \circ B_2 \in \mathbb L(Z,Y)$ compact? (Where $B_1 \...
0
votes
0answers
27 views

Banach-Steinhaus Theorem - “Shift”.

How do I find? By theorem we know that: (Banach-Steinhaus Theorem). If $(T_n)_{n=1} ^\infty$ is a sequence $B(\mathcal{B} , \mathcal{N})$ so that, for all $\xi \in \mathcal{B}$, there is a limit $T \...
0
votes
0answers
16 views

isometric identification of Banach sapces

I saw a conclusion:$E,F$ are Banach spaces,$S$ is the set of all bounded linear maps from $E$ to $F^*$,$T$ is the set of all bounded bounded forms on $E\times F$.How to show $S$ is isometric ...
2
votes
1answer
35 views

Uniform boundedness theorem.

Let $V$ be subspace of $\ell^2$ which contains all 1 summable sequences. For each natural number $n$, define $T_n: V \to \mathbb R$ by $T_n(x)=\sum_{i=1}^n x_i$. Then $T_n$ is not uniformly bounded on ...
2
votes
0answers
26 views

Group of invertible isometries on $\ell^p$

I know that the group of unitary operators on infinite dimensional separable Hilbert space is connected but I would like to know whether the group of invertible isometries on $\ell^p$ is connected ...
0
votes
0answers
17 views

Compactness of linear mapping from a Banach space to another

Let X, Y be Banach spaces and $ X \hookrightarrow Y$ (which would imply $Y' \hookrightarrow X'$). Let the bilinear form $a_0: X \times X \to \mathbb R$ be continuous, symmetric and coercive. Moreover, ...
1
vote
1answer
33 views

Operator into the dual space is compact

I want to solve the following Let $X,Y$ be Banach spaces, with compact embedding $X\hookrightarrow Y$. Define the bilinear form $b:X\times Y\to\mathbb{R}$ that satisfy $$b(u,v)\leq C\|u\|_X \|v\|...
0
votes
1answer
26 views

prove uniqueness Lipschitz-continuous map in Banach space

Let $X$ be a Banach space and let $T:X \rightarrow X$ be a Lipschitz-continuous map. Show that for $\mu$ sufficiently large, the equation $$Tx + \mu x = y$$ has, for any $y \in X$, a unique solution. ...
0
votes
0answers
26 views

Cauchy-Lipschitz-Picard theorem in Banach spaces

In his famous book, just before proving Hille-Yosida theorem, Brezis gives a generalization of the Cauchy-Lipschitz theorem for existence and unicity in Banach spaces, which I will call $E$. At the ...
1
vote
1answer
42 views

What is the reason for considering a Banach space in the following theorem?

Theorem $:$ Let $(X,\|\cdot\|)$ be a Banach space. Let $T:X \longrightarrow X$ be a continuous linear transformation. Suppose $a=\|T\| < 1$. Then there exists $S:X \longrightarrow X$ such that $ST=...
0
votes
0answers
23 views

Does the completion of an operator with closed image also have a closed image?

Let $X,Y$ be real normed vector spaces, and suppose that $T:X \to Y$ is a bounded linear operator with closed image. Let $\tilde X,\tilde Y$ be the completions of $X,Y$, and let $\tilde T:\tilde X \...
3
votes
0answers
41 views

Banach spaces and compactness

Let X, Y be Banach spaces and $ X \hookrightarrow Y$ (which would imply $Y' \hookrightarrow X'$). Let the bilinear form $a_0: X \times X \to \mathbb R$ be continuous, symmetric and coercive. Moreover, ...
0
votes
2answers
31 views

A set under multiplication by non-negative real numbers in Banach space

Let $X$ be a Banach space and $S$ a closed subset of $X$. Is the set $\tilde S=\{rs;\ s\in S,\ r\geq 0\}$ also closed?
1
vote
1answer
39 views

Proof of a criterion for Bochner integrability

The following comes from the Wikipedia article on Bochner integrals. Theorem. If $(X,\Sigma,\mu)$ is a measure space, then a Bochner-measurable function $f:X\to B$ is Bochner integrable if and only ...
2
votes
0answers
32 views

Soft quesion: what do we lose if we assume measures are complete and $\sigma$-finite in integration theory?

This is intended to be a really soft question, which might be considered bad for this site. Let me know if that's true. I'll try to make this question as "answerable" as possible. I've been reading ...
1
vote
0answers
32 views

Natural tramsformation between a normed linear space and it's double dual.

Let $X$ be a normed linear space. Then I know that $X$ is imbedded in it's double dual $X^{**}$ via the natural tramsformation $J : X \longrightarrow X^{**}$ (say). I have proved that $J$ is one-to-...
3
votes
1answer
49 views

To prove $(e_n-e_{n-1})$ is basis for $X$

Let $(e_n)$ be a normalized basis for a Banach space $X$ and suppose there exists $x^{*} \in X^{*}$ with $x^{*}(e_n)=1$ for all n. Show that the sequence $(e_n-e_{n-1})_{n=1}^{\infty}$ is also a basis ...
1
vote
1answer
28 views

Dual of the product is isometric to the product of the dual

Let $X,Y$ be Banach spaces. Let $Z = X\times Y$ equipped with the $p$-norm, where $||(x,y)||_Z = (||x||_X^p + ||y||_Y^p)^{1/p}$. Suppose $X^* \times Y^*$is equipped with the $q$-norm where $1/p+1/q =...
0
votes
0answers
30 views

Hahn Banach extensions.

Consider the normed linear space $R^2$ equipped with the norm given by $||(x,y)||=|x|+|y|$ and the subspace $X=\{(x,y)\in R^2 : x=y\}.$ Let $f(x,y)=3x$ on $X$. Then what is the Hahn Banach extension ...
1
vote
0answers
18 views

Operator in funtional space

Let $X = \{f :[0,1] \rightarrow \mathbb{R}/ f \hspace{1.0mm} is \hspace{1.0mm} continous\}$ and $T(f)_{(x)} = \int_{0}^{x}{f_{(s)}ds} $. Prove that $\hspace{3.0mm}T^n(f)_{(x)} = \int_{0}^{x}{K_{n}(x,...
0
votes
0answers
21 views

Proving that $l_∞$ is complete

I am using the theorem : An NLS $V$ is complete if and only if every absolutely summable series converges in $V$ Let $\sum x^n$ be absolutely summable where $x^n:=(\xi^n_1,\xi^n_2,...)$. Let $s^n:...
0
votes
0answers
31 views

Convert Real banach space to complex banach space

Suppose I have a real vector space $V$ and I would like to extend the scalar multiplication in such a way that I obtain a complex vector space. It is not difficult to see that doing so is equivalent ...
0
votes
0answers
21 views

It is banach space or just normed space ? [duplicate]

Let $X$ be a nonempty set and $B(X,C)$ the set of all complex functions defined on $X$ and $\sup|f(x)|<+\infty$. Define norm $||\cdot||$ on $B(X,C)$ by $||f||=\sup|f(x)|$. Is $(B(X,C),||\cdot||)$ ...
1
vote
1answer
28 views

About dense subspaces in Banach spaces

I have the following problem: Let $E$ a Banach space and $X_1,X_2$ dense subspaces. Is $X_1\cap X_2$ dense in $X$? What is the answer if $X_1,X_2$ has codimension 1? I don't know how to start. If ...
0
votes
0answers
40 views

Perplexing integral equation

In the proof of Theorem 2.16 (a generalized chain rule) in the article A general fredholm theory i: a splicing-based differential geometry by Helmut W Hofer, Krzysztof Wysocki, and Eduard Zehnder, ...
0
votes
1answer
32 views

Continuous linear functional on $c_{00}$

Let $X$ be the space of real sequences having finitely many nonzero terms with $||\ ||_p , 1\leq p \leq \infty$ . Define $f: X\to\mathbb{R}$ by $f(x)=\sum_{j=1}^{\infty} x_j $ for $x=(x_j) \in X$. ...
1
vote
0answers
10 views

Can a non-invertible isometry on $L^p$ space have index zero?

When $p=2$, a non-unitary isometry on Hilbert space is unitarily equivalent to a unilateral shift so it cannot have Fredholm index zero. When $p\in[1,\infty)\setminus\{2\}$, there is a version of ...
1
vote
1answer
47 views

Question about $\ell^{p}$ spaces

I am quite new to the subject of sequence spaces. I got a few doubts (hope they are not silly). While reading about $\ell^{p}$ spaces, I read that these spaces equipt with the $p$-norm form normed ...
4
votes
1answer
64 views

Explicit exemple of a derivative which changes when we change the norm

I know that if $E$ and $F$ are finite-dimensional Banach spaces and $f:E\to F$ is differentiable, then $\mathrm{d}f$ does not depend on the choice of norms in $E$ and $F$ (since all the norms are ...
0
votes
0answers
25 views

Please check my proof of completeness for a sequence space, and make it concise.

Let $\mathcal{O}(\mathbb{C}) := \{(c_n)_{n \in \mathbb{N}}: \sum_{n=0}^{\infty} r^n |c_n| \le \infty \;\text{for all}\; r > 0 \}$. The directed set of semi-norms $\|(c_n)\|_r := \sum_{n=0}^{\...