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

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Question related to differentiable functions on Banach spaces

There is an interesting exercise on my Analysis book that I have not been able to solve: Let $\mathbb{E,F}$ be Banach spaces, $f:\mathbb{E}\to\mathbb{F}$ of type $\mathcal{C}^k$, $k\geq1$. Asume $f'(...
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Show that $(L^{p},\|\|_{p})$ is a Banach space.

Show that $(L^{p},\|\|_{p})$ is a Banach space. My approach: I prove the statement for $(L^{1},\|\|_{1})$, of the following way, first all, is easy show that $\|\|_{1}$ is a norm. So, $(L^{1},\|\|_{...
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32 views

am I misunderstanding boundness of a transformation?

Let $X$ be a Banach's space $Y$ a normed vector space $H\in B(X,Y)$ a family of bounded linear transformations $X\rightarrow Y$ and $V_n:=\{x\in X:\exists T\in H$ such that $||Tx||>n \} n\in \...
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43 views

Sum of closed spaces is not closed

I recently encountered the theorem that the sum of a closed (linear) subspace with a finite dimensional subspace is closed subspace of the Banach space in which it is contained. However, this came ...
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37 views

Sequences - Convolution

We got the following setting: Let $A$ the set of all complex sequences $x=(x_n)_{n \in \mathbb N_0}$ with \begin{align*} \Vert x \Vert := \sum_{n = 0}^\infty \vert x_n \vert e^{-n^2} < \infty. \...
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$\bigcap_{S \in L(E,F)} ker(S) = \{0\}$

Let $E$ a Banach space, $F$ a normed space and $L(E,F)$ a set of bounded linear operator from E to $F$. Is true that $$\bigcap_{S \in L(E,F)} ker(S) = \bigcap_{S \in L(E,F)} S^{-1}(0) = \{0\}.$$ If ...
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A question concerning Mazur's Lemma

I have a problem with application of Mazur's Lemma. Just consider $B(H)$ when $H=\ell_2$. Then, $B(H)$ is a normed vector space. Then, take operators $$X_n:={\rm diag}(0,0,\cdots,0,1,1,1,1,\cdots)$$...
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22 views

The space $C_0$ does not have a 2-dimensional subspace isometric to $E^2$?

To prove that the space $C_0$ of the sequences x=($x_1,x_2$,...) for which $\displaystyle \lim_{i=\infty} x_i=0$ with the norm $\|x\|=\max_{1 \leq i < \infty} |x_i|$ does not have a two-dimensional ...
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41 views

Schauder bases of subspaces of the sequence space $\ell^p(\mathbb{N})$

Consider the canonical Schauder basis $\{e_i:i\in \mathbb{N}\}$ for $\ell^p(\mathbb{N})$, where $e_i(j)=\delta_{ij}$. Let $M$ be a subspace of $\ell^p(\mathbb{N})$. Is it right that $\{e_i:i\in \...
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counterexample for $C^1(U)$ not complete in any dimension

Cleary $C^1[a,b]$ is not complete with $\|\cdot\|_{\sup}$. I am looking for a counterexample which is working in any dimension, i.e. $C^1(U)$ is not complete for any open $U\subseteq \mathbb R^n$ ...
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22 views

Interpolation of a subspace of codimension one

I am a little bit lost in interpolation theory. Let $A$ be a linear operator on $\mathbb{R}^{n}.$ Denote $V_0$ a subspace of $\mathbb{R}^n$ of codimension one. Suppose that $AV_0 \subseteq V_0$ and we ...
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52 views

Weak closure of the boundary of the unit ball of $l^1$

I am trying to prove that zero is in the weak closure of set $\{ x\in l^1 \,:\, \lVert x\rVert =1\}$ . And I need example too. Is there any countable subset of this set which has $0$ in weak closure?
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29 views

Convergence of series of linear operators in Banach space

Let $X$ be a Banach space and $A\in L(X,X)$. Show that $$\sum_{k=0}^\infty \frac{A^k}{k!} $$ converges in $L(X,X)$. Find an upper bound to the norm of sum. If a series converges absolutely, then it ...
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27 views

Existence (and construction) of a convergent series

Suppose $(b_n)$ is an unbounded (real or complex) sequence. Does there always exist some (absolutely) convergent series $\sum a_n$ such that $\sum |a_nb_n|$ (or better, $\sum a_nb_n$) diverges? If so, ...
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38 views

the “real spectrum” of an operator acting on a real Banach space

Let $X$ be a Banach space over the field $\mathbb{R}$, and denote by $\mathcal{L}(X)$ the space of continuous linear operators acting on $X$. The spectrum $\sigma(T)$ of an operator $T\in\mathcal{L}(...
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23 views

Show that there is a unique function on the interval that solves:

$$f(x) = e^{-2x} + \int_0^\infty e^{-2x-2y} \sin(x-y)f(y) \, dy$$ I can't get a good bound on $e^{-2x-2y} \sin(x-y)$ so that I can apply Banach.
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45 views

Continuity on [0,2) of the modulus of convexity

For a Banach space $X$ the mapping $$\delta_X\colon[0,2]\to[0,1], \varepsilon\mapsto \inf\{1-|x+y|/2\colon |x|\leq 1, |y|\leq 1, \text{and } |x-y|\geq\varepsilon\} $$ is called the modulus of ...
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A uniformly continuous on the unit sphere of $c_0$ is bounded

Let $S_{c_0}= \{ x\in c_0 \;: \Vert x \Vert =\;1\; \}$, where $c_0$ is the space of all sequences converging to zero and $$f: S_{c_0} \rightarrow \mathbb R$$ is a uniformly continuous function. Prove ...
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spectrum of a unilateral left weighted shift operator on c0

The spectrum of shift operators on $\ell_p$ is well-documented, however the spectrum of shift operators on $c_0$ does not seem to be nearly as commonly discussed. Definitions. Let $(e_n)_{n=1}^\...
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Not unique Hahn Banach extension

$G = \{(x_n) \in l_1: x_{2n+1} = 0, \forall n \in \mathbb{N} \}$ Let $f: G \to \mathbb{K}$ be a continuous linear functional, $f \neq 0$. Show that the Hahn Banach extension of $f$ is not unique. ...
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Uniqueness of a Hahn Banach extension

Let $M = \{(x_n) \in l_1: x_1-3x_2=0\}$ and $f: M \to \mathbb{R}$ given by $f((x_n)) = x_1$. Let $g: l_1 \to \mathbb{R}$ be the Hahn Banach extension of $f$. Show that $g$ is unique. I know that $\...
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Problem. Convergence. Banach space. Weak topology

Let E be a Banach space and let $(X_n)$ be a sequence such that $X_n \rightharpoonup x$ in the weak topology σ(E,E'). Set: $S_n=\frac{1}{n}\sum_{k=1}^n(-1)^kX_k$ Does $Sn \rightharpoonup x $ in the ...
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Banach-Mazur distance from finite-dimensional subspaces of $\ell_p$ to the Hilbert space

I am reading a paper http://www.math.tamu.edu/~johnson/TF3.4.pdf by Bill Johnson and Andrzej Szankowski and having trouble grasping why $d_n(Z_m) \leq d_n(\ell_{p_{m+1}} ) = n^{|p_{m+1}-2|}$ in the ...
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43 views

A not so obvious corollary?

Let $X$ be a normed vector space over $\mathbb{K}$. Then there is only one completion of $X$, the banach space $\hat{X}$ such that $X$ is a dense subspace of $\hat{X}$. I am trying to prove that there ...
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Isometric embedding of $\ell ^ 1$ in $\ell ^\infty$ in finite dimensions

It is well known that $\mathbb{R}^n$ with $\ell ^1$ norm can be embedded into $\mathbb{R}^k$ with $\ell^\infty$ norm for some $k\in \mathbb{N}.$ But I guess, this is not true in complex case that is ...
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Banach Theorem on Metric Space for Integral Equations

My instructor said that Banach doesn't apply in this case: f(x) = sin(x) + $\int_0^x$$f^2$(z)dz f(0) = 0; f'(0) = 1 > 0 f'(x) = cos(x) +$f(x)^2$, which is positive on (0,$\pi$) so f is positive ...
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42 views

Is there a Schauder basis of the Banach space $l_p(l_q)$

For Banach space $l_p(l_q)$, where $p,q\geq 1$, does the following vectors form a Schauder basis of it, say a rearrangement of all $v_{m,n}=(0,...,0,e_m,0,...)$, where $e_m$ occur in $n$th coordinate ...
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62 views

Prove that $(a_n) \in l_2$.

Suppose that $\sum_{k=1}^{\infty} a_k x_k$ converges for all $x=(x_n)\in l_2$. Prove that $(a_n) \in l_2$. My attempt Let $T_n: l_2 \to \mathbb{K}$, $T_n(x) = \sum_{k=1}^{n} a_k x_k$. ($\mathbb{K} =...
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67 views

A possible norm on a subspace of $C^\infty([0,1])$?

My question is related to this one: Take the vector space of infinitely differentiable functions on $[0,1]$. The standard norm of $C^k([0,1])$ is just the $\ell^1$-norm of the vector $(\|f\|_\infty, \|...
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Equivalences for a Banach lattice

I'm trying to prove the following equivalences for a Banach lattice $E$: $E$ has an order continuous norm Every monotone order bounded sequence in $E$ is convergent E is an ideal in $E^{**}$...
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Does the pth James space Jp contain a norm-1 basic sequence domimated by l2? Equivalently, is there a noncompact operator from l2 into Jp?

The $p$th James space, denoted $J_p$, is just the regular James space using the $p$-norm in place of the 2-norm. See here for a complete definition. To use their notation, let $\mathbb{N}_0$ denote ...
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45 views

weak convergence and weak * convergence criterium

I have to solve the following problem Let $X$ be a Banach space. Prove that $x_n\rightharpoonup x$ in $X$ if and only if $\sup||x_n||<+\infty$ there exists a dense subset $E'$ of $X'...
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A confusion about the norm of the restriction of a linear mapping.

Let $\Bbb X$ be a Banach space, $T:\Bbb X\to \Bbb X$ be a linear map and $P:\Bbb X\to \Bbb X$ be a projection operator. Denote the closed subspace that is the range of $P$ by $\Bbb Y:=\mathcal R(P)$. ...
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32 views

Finite Dimensional Hilbert Space

A while ago someone asked this question. I really like what the accepted answer is trying to do. But, I am having trouble figuring out his justification for the first line in the proof: $$\bigcup_{x \...
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75 views

Why does this proof work: Closed unit ball in $C_0$ is not compact

I know that this question has been asked to death, and multiple solutions are given, but I still don't understand why the "standard" proof works Following Show that the closed unit ball $B[0,1]$ in $...
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44 views

Hahn Banach Theorem Application

I want to proof that exists $f \in l_\infty '$ with $f(x) = \lim x_n, \forall x = (x_n) \in c$ and $f(x_1, x_2, x_3,...) = f(x_2,x_3,x_4,...)$ What I have been doing until now: Consider the ...
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About a particular linear map between sequence spaces

Let $x \in \ell^1$ and $z \in \ell^2$ taking values in $\mathbb{R}$ and define a linear map $T_z: \ell^1 \rightarrow \ell^2$ as follows: $y_1=0$ and $y_n=\sum_{k=1}^{n-1}z_{n-k}x_k$ for $n\geq 2$. ...
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Weakly convergent series

Consider sequence $\{x_n\}_{n = 0}^\infty $ in complex Banach space X. Suppose that for all $z \in \mathbb{C}$ such that $|z| < 1$ the series $\sum_{n = 0}^\infty z^nx_n$ is weakly convergent. How ...
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Problem following proof of Šmulian theorem for separable space

I tried to solve Problem 10 on p. 464 of Brezis to get a proof of part of the Eberlein-Šmulian theorem, precisely the equivalence between compactness and sequential compactness in the weak topology of ...
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Is it true: If all linear subspaces of a Banach space are closed, then the space is of finite dimensions?

Is it true: If all linear subspaces of a Banach space are closed, then the space is of finite dimensions? My attempt to prove this: For contradiction, suppose $X$ is an infinite-dimensional ...
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Sequence of bounded Operators (Is this a counterexample?)

I've to proof the following statement Let $X,Y$ be to banach spaces and $(T_k)_{k \in \mathbb{N}} \subseteq L(X,Y)$ a bounded sequence of bounded linear operators. Further it exists a dense subset ...
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32 views

Banach space and invertible linear operator

Let $X$ be a Banach space. Let $T : X \to X$ be a invertible linear operator and $M > 0$ be such that $\|T^{-k}\| \le M$ for all $k \ge 1$. Prove that $\inf_ {n\ge1} \|T^n(x)\| > 0$ for all $x \...
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54 views

Proof, that a set is not convex

I try to solve the problem 106 of the scottich book. I know the set of all rearranged sums is convex. Let $f_{j,k}$ the indicator function of the interval $(\frac{j}{2^k},\frac{j+1}{2^k}$). k = 0,1,2,...
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52 views

Symmetric norms on $\mathbb{R}^n$ are between the $\ell^1$ and $\ell^\infty$ norms

We say that a norm $\|\cdot\|$ on $\mathbb{R}^n$ is symmetric if it is invariant under sign changes and permutations of the components. Suppose that $$\|(1, 0, ..., 0)\| = 1$$ for a symmetric norm. ...
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completion of $C^{\infty}\left(S\right)$ is $L^2(S)$?

I have the space of infinitely derivable functions, i.e. $C^{\infty}\left(S\right)$ with the following inner product $$\left\langle f,g\right\rangle =\intop_{0}^{2\pi}\intop_{0}^{2\pi}\left(\overline{...
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494 views

Can the real vector space of all real sequences be normed so that it is complete ?

Let $X$ be the vector space of all real sequences . Does there exist a norm on $X$ which makes it complete ?
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22 views

Affine contractions from linear contractions?

Let $V$ be a linear space. Consider a contractive linear map $M:V\mapsto V$, $$ \|Mv\|\leq \|v\| \quad \text{for all vectors } v\in V. $$ Now, for some fixed vector $c\in V$, the question is to sort ...
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$X$ be finite dimensional real NLS , let $x \in X$ , does there exist $T \in \mathcal B(X)$ such that $\{T^n(x):n \in \mathbb N\}$ is dense in $X$?

Let $X$ be a finite dimensional real normed linear space , let $x \in X$ , then does there exist a continuous linear transformation $T:X \to X$ such that $\{T^n(x):n \in \mathbb N\}$ is dense in $X$ ?
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$f \in \mathcal l^{\infty}{'} $ ; $f(x)\ge 0$ whenever $x \in \mathcal l^{\infty} $ is a sequence of non-negative terms ; is $f$ bounded? [duplicate]

Let $f:\mathcal l^{\infty} \to \mathbb R$ be a linear functional such that $f(x)\ge 0$ whenever $x \in \mathcal l^{\infty} $ is a sequence with non-negative terms ; then is $f$ continuous ?
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isomorphism between $C[0,1]$ and $C^1[0,1]$

Is space $C[0,1]$ with norm $\parallel f \parallel=\max|f(x)|$ (space of continuous functions on $[0,1]$) isomorphic to space $C^1[0,1]$ with norm $\parallel f \parallel=\max|f(x)|+\max|f'(x)|$ (space ...