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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Showing that $\lim \int \left(\sum_1^n |f_k|\right)^p \le \left(\sum_1^\infty \|f_k\|_p\right)^p$

I am reviewing a proof about the completeness of $L^p$ spaces. The proof begins as such (Folland Theorem 6.6): For $1 \le p < \infty$, suppose $\{f_k\} \subset L^p$ and $\sum_1^\infty \|f_k\| = ...
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0answers
6 views

If a Banach space is a conjugate space and have an unconditional basis, then is weakly sequentially complete?

It is well known that a weakly sequentially complete Banach space with an unconditional basis is isomorphic to a conjugate space. Is the inverse of this statement true? If a Banach space is a ...
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1answer
33 views

Prove: $M +N$ is closed in $B$ iff $∥m∥_B +∥n∥_B ≤ c∥m+n∥_B$, for all $m ∈ M$ and $n ∈ N$.

Let $B$ be a Banach space and $M,N$ closed subspaces of $B$ such that $M ∩N = \{0\}$. Prove: $M +N$ is closed in $B$ iff $∥m∥_B +∥n∥_B ≤ c∥m+n∥_B$, for all $m ∈ M$ and $n ∈ N$. My Work: If ...
3
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1answer
73 views

I am trying to find a Banach Space X and a singular operator (Infinite Dimensional Space)

I am trying to find a Banach space $X$ (Infinite dimensional Space) and a singular operator $A\in \mathcal L(X) $ such that for some $\epsilon \gt 0, $ there is no bounded linear operator $B$ with ...
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1answer
32 views

Given a singular matrix, I am tring to find an invertible matrix… (Finite Dimensional Space)

In coordinates and in a finite-dimensional space, how would I prove that given any singular $n$x$n$ matrix $A$, any $\epsilon\gt0$ and any matrix norm $||.||$, there is an invertible $n$x$n$ matrix ...
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0answers
7 views

Are two operator norms on $M_n(A)$ equivalent?

If $A$ is a Banach algebra, then $M_n(A)$ can be given the operator norm as operators on $A\oplus_p\cdots\oplus_p A$ ($1<p<\infty$) to make it a Banach algebra. If in addition $A$ is an operator ...
1
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1answer
50 views

$(B_X,w)$ metrizable implies $X^\ast$ separable

Let X be a normed space and assume that $(B_X,w)$ is metrizable, i.e. the weak topology is metrizable. Show that $X^\ast$ is separable. My attempt: Let $d$ a equivalent metric on $B_X$. For fixed ...
0
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1answer
17 views

check if a linear operator is bounded

show that $Tf = f(0)$ is not a bounded linear functional on the space of continuous functions measured with the L2 norm, but it is a bounded linear functional if measured using the uniform norm. ...
2
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1answer
83 views

Proving that $c$ is a Banach space.

I want to prove that $$c = \{ (x_n)_{n\geq 0} \mid x_n \in \Bbb C \text{ and the sequence converges} \}$$ with the norm $$ \left\|(x_n)_{n\geq 0}\right\|_{\infty} =\sup_{n\geq 0} |x_n|$$ is a Banach ...
4
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2answers
44 views

Checking that $(C[0,1], \|\cdot\|_1)$ is not Banach.

I want to check that $(C[0, 1], \|\cdot\|_1)$ is not a Banach space, where $$\|f\|_1 = \int_0^1 |f(x)|\,{\rm d}x.$$ I took $(f_n)_{n \geq 1}$ a sequence in $C[0, 1]$ given by: $$f_n(x) = ...
4
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1answer
2k views

About Banach Spaces And Absolute Convergence Of Series

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, ...
4
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2answers
103 views

Any relations between the weak topology on a Banach Space and the weak topology on CW complexes?

I'm learning about CW complexes, which we'll say are topological spaces $X$ that admit a filtration $\emptyset \subset X^0 \subset X^1 \subset \cdots \subset X^n \subset \cdots$, with $X=\bigcup X^n$, ...
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1answer
14 views

Definition of finite dimensional decomposition of Banach space

The question is in the title. What is the definition of finite dimensional decomposition of Banach space? I have been looking around for a while and can't find anything! Thanks
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1answer
251 views

Are there spaces “smaller” than $c_0$ whose dual is $\ell^1$?

It is well known that both the sequence spaces $c$ and $c_0$ have duals which are isometrically isomorphic to $\ell^1$. Now, $c_0$ is a subspace of $c$. My question - is there an even smaller subspace ...
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1answer
320 views

Definition of Frechet derivative

Let $f: U \to Y$ be a map from $U \subset X$ open, $X,Y$ Banach spaces. Then the Frechet derivative at $x_0 \in X$ is defined as follows: $f$ is called Frechet differentiable at $x_0$ if there ...
0
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3answers
55 views

show that $l^2$ is a Hilbert space

Let $l^2$ be the space of square summable sequences with the inner product $\langle x,y\rangle=\sum_\limits{i=1}^\infty x_iy_i$. (a) show that $l^2$ is H Hilbert space. To show that it's a ...
7
votes
2answers
100 views

The spectrum of a self-adjoint operator on $\mathcal l^2$

Let $S$ be the unilateral shift operator on $\mathcal l^2$ (which shifts one place to the right) and $S^*$ its adjoint, the backward shift (which shifts one place to the left). I've been trying to ...
2
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0answers
24 views

Open problems in Banach spaces? universal spaces

I have gathered a list of universality problems in Banach spaces which have been solved: The non existence of a separable reflexive space universal for the class of separable reflexive spaces. If a ...
2
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1answer
44 views

Completeness of $C^1$ functions vanishing at infinity with sup-norm of derivatives

I'm looking at $$C_0^1(\mathbb{R}) := \{f \in C^1(\mathbb{R}) : \lim_\limits{|x|\rightarrow \infty}f(x) = \lim\limits_{|x|\rightarrow \infty} f'(x) = 0\},$$ along with the norm given by $||f|| := ...
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2answers
27 views

Correspondence between linear maps of a vector space into itself and linear maps of the dual into itself.

I was wondering about vector spaces and their dual. Specifically, in the context of finite-dimensional vector spaces, I asked myself if it is true that there is a one-to-one correspondence between the ...
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2answers
49 views

Why is it important that $L^P$ spaces be complete?

I know that Banach spaces are ubiquitous and incredibly important in a lot of areas of math, but I was hoping for an intuitive explanation as to why (and when) it's important in the case of $L^p$ ...
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1answer
96 views

Is the complex Banach space $C([0,1])$ dual to any Banach Space?

I've been able to show that the extreme points of $C([0,1])$ are the continuous functions that take values on the unit circle. However, I'm not sure how to reason from here as to whether or not it is ...
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1answer
152 views

Banach Spaces: Uniform Integral vs. Riemann Integral

Problem Given a finite measure space $\Omega$ and a Banach space $E$. One has strict inclusion: ...
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1answer
33 views

Show space C([0,1]) with norm integral is a Banach space [duplicate]

Is the space C([0,1]) with the norm integral from 0 to 1 of |f(t)|dt a Banach space?
3
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1answer
165 views

Showing that the space of Hilbert-Schmidt operators form a Banach space.

How do i show that the set of Hilbert-Schmidt operators $HS(H) = \{T \in B(H) \; : \; \sum^{\infty}_{n=1}\|Te_n\|^2 < \infty \}$ for some countable ONB $\{e_n\}$, on a separable Hilbert Space ...
1
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1answer
46 views

$\lambda f(x)+(1-\lambda)f(y)=f(\lambda x+(1-\lambda)y)$ implies $f $ linear?

Let $X$ be a Banach space. $f:X \to X$ a continuous function. If we assume that $f$ satisfies the following convexity condition: $$\lambda f(x)+(1-\lambda)f(y)=f(\lambda x+(1-\lambda)y),$$ for all ...
2
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0answers
41 views

Let $V:= (C([0,1]),\|\cdot\|$) with $\|f\|:= \int_0^1|f(x)| \ dx$. Consider the function $f_n$ and show that V is not a Banach Space.

The Assignment: Let $V:= (C([0,1]),\|\cdot\|$) with $\|f\|:= \int_0^1|f(x)| \ dx$. Consider the continuous function $f$ for all $n \in \mathbb{N}$: $$f_n: [0,1] \rightarrow \mathbb{R}, \ ...
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1answer
11 views

Is the completion of a separable normed linear space is also separable?

Let $X$ be a separable normed linear space. Is the completion of $X$ is a separable Banach space?
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0answers
57 views

Another proof of Inverse Function theorem in $\mathbb{R}$

(Inverse Function theorem in $\mathbb{R}$) Suppose $I\subset \mathbb{R}$ is an open interval and $f:I\rightarrow\mathbb{R}$ is a differentiable function.If for all $x\in I$ is such that $f^{'}(x)\ne ...
0
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2answers
39 views

Extreme points of unit ball of Banach spaces $\ell_1$, $c_0$, $\ell_\infty$

Find extreme points of the unit balls of each Banach space, $l^1 $, $c_0$, $ l^\infty$ Can you help me with this one? For the first space, $l^1$, I thought there was no extreme point, but ...
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0answers
13 views

Explicit example of tensor norms

I can't find any example anywhere on the web where someone actually evaluates a non-trivial tensor norm. So I'm wondering about the simplest non-trivial case. Let $X$ be $\mathbb R^2$ with the ...
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1answer
23 views

Adjoint operator in Banach space

From Functional analysis, by Conway. I try to prove this exercise. If $ X $ and $ Y $ are Banach spaces and $ B \in \mathscr B(Y^*, X^*) $, then there is an operator $ A $ in $ \mathscr B(X,Y) $ ...
2
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0answers
25 views

Extreme points of the unit balls of $l^\infty, C([0,1])$

Determine the extreme points of the unit balls of $l^\infty$, and $C([0,1])$ for real-valued functions, with the uniform norm. Is $C([0,1])$ the dual of a Banach space? I've found the extreme points ...
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3answers
37 views

Surjectivity of $Id-A$ for linear operator $A$ on Banach space with $\|A\|<1$

Let $X$ be Banach space and $A:X\rightarrow X$ linear opeartor such that $\|A\|<1$. It is clear that $Id-A$ is injective. Why is it also surjective?
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15 views

Let $E$ be the space $L^1(\mathbb{R})\cap L^2 (\mathbb{R})$ equipped with the norm $\|u\|_E = \|u\|_1 + \|u\|_2$.

I am trying to solve this but I got stuck. Help needed. $E$ is a Bananch space. Let $f(x) = f_1(x) + f_2(x)$ with $f_1\in L^\infty(\mathbb{R})$ and $f_2 \in L^2(\mathbb{R})$. Check that the mapping ...
3
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0answers
28 views

Continuity of a function in a locally convex topological space

I endow the space of bounded sequences with a locally convex topology $\tau$ such that $\tau$ is strictly finer than the product topology (the topology of pointwise convergence), $\tau_p$, and ...
2
votes
1answer
27 views

Adjoint operator on Banach space

Suppose $X$ and $Y$ are Banach spaces and $T:X\to Y$ is a bounded linear operator. Show that $T$ is an isometric isomorphism if and only if its adjoint $T^*$ is also an isometric isomorphism. Given an ...
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1answer
37 views

Bounded Operator Norm: Special Element

Given a Banach spaces $X$ and $Y$. Consider a bounded operator: $$T:X\to Y:\quad\|T\|<\infty$$ Then theres an element: $$\|Tx\|=\|T\|\cdot\|x\|\quad(x\neq0)$$ Does it always exist?
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1answer
12 views

Exercise about adjoint of densily defined operator between Banach spaces

Let $X$ and $Y$ be Banach spaces and $A:D(A)\subset X \to Y$ a densily defined linear operator. Suppose the graph of $A$ is closed. Then the follwing are equivalent: $D(A)=X$; $A$ is bounded; ...
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1answer
63 views

When is the dual ball of $L_1(\mu)$ weak*-sequentially compact?

Where could I find a direct proof showing that the dual ball of $L_1(\mu)$ is weak*-sequentially compact? Since $(L_1(\mu))^*=L_\infty(\mu)$, I mean the unit ball $B_{L_\infty(\mu)}$ of ...
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0answers
66 views

Does a Banach valued Cauchy series over arbitrary set converges?

Definitions: If $f$ is a function from a set $A$ (not necessarily countable) into a Banach space $V$, we say that the series of $f$ over $A$ converges if there exists an element $v \in V$ such that ...
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26 views

Is image of sum of maps closed?

Suppose $V$, $W$ are Banach spaces and $f$, $g:V \rightarrow W$ are continuous linear operators and $\operatorname{Im}f$, $\operatorname{Im}g$ are closed subspaces of $W$. Is it true that ...
131
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4k views

Does the open mapping theorem imply the Baire category theorem?

A nice observation by C.E. Blair1, 2, 3 shows that the Baire category theorem for complete metric spaces is equivalent to the axiom of (countable) dependent choice. On the other hand, the three ...
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2answers
32 views

$L^p$ spaces and proper inclusion

Let $1≤p < q$. Prove that $L^p(\mathbb{R}) \subset L^q(\mathbb{R})$ and the inclusion is proper. I am unsure how to begin this or even prove it about $L^p$ spaces and Banach spaces.
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1answer
30 views

Does every closed subspace of a dual space correspond to a closed subspace of its predual?

Suppose $X$ is a Banach space with dual space $X^*$. If $Y$ is a closed subspace of $X$, then $Y^\perp=\{x^*\in X^*: x^*(y)=0 \text{ for all } y\in Y\}$ is a closed subspace in $X^*$. I am wondering ...
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1answer
23 views

Analytic sets,, effros borel structure

Let SB denote the set of closed subspaces of $C(2^\mathbb{N})$ equipped with the Effros Borel structure, and $A\subset$ SB be analytic. I am reading a proof that says $A_\sim = \{Z\in $SB $ \mid $ ...
0
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1answer
22 views

Isomorphism, Separable spaces

I am trying to show that: If there are sequences $(x_n)\subset X$ and $(y_n)\subset Y$ where $X$ and $Y$ are separable Banach spaces, such that $\overline{sp}\{x_n \mid n\in\mathbb{N}\}=X$ and ...
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1answer
30 views

Closed set in Baire space

I am reading a book on Banach spaces. It introduces the Baire space $\mathcal{N}=\mathbb{N}^\mathbb{N}$ as the product of infinitely many copies of $\mathbb{N}$ with the discrete topology. We have ...
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1answer
38 views

Is the relative interior of a subspace which is not closed empty?

In a general Banach space, the relative interior of a linear subspace which is not closed is empty, why ?
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1answer
20 views

Strong convergence of product of operators on a Banach space

If $\{T_n\},\{S_n\}$ are two sequences of bounded operators on a Banach space $X$, such that $\{T_n\}$ converges weakly to $T$, and $\{S_n\}$ converges strongly to $S$, does it follow that $T_nS_n\to ...