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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Application of the Banach fixed point theorem

Let $a > 0$. We consider the function: $f: (0, \infty) \to (0, \infty)$, defined by $f(x) = \frac{1}{2}(x + \frac{a}{x})$. Let $(x_n)_{n \in \mathbb{N}_0}$ be defined by: $x_0 \in (0, \infty)$, ...
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83 views

A copy of $l_\infty$ in a infinite dimensional Banach space

Let $E$ an infinite dimensional Banach space. Using the Hahn-Banach extension theorem, prove that there is a sequence $(y_n)\subset E$ and a decreasing sequence of closed subspaces ...
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44 views

The property of closed subspace

We know that a set is closed if and only if every convergent sequence with elements in the set has a limit point in the set. I am reading a paper, and the paper claims that the following is due to S ...
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49 views

Show that $B(X)=\{f:X \rightarrow \mathbb{R}: f \text{ is bounded}\}$ is complete.

Given a metric space $(X,d)$ consider the metric space $B(X)=\{f:X \rightarrow \mathbb{R}: f \text{ is bounded}\}$ with the distance $d_{\infty}(f,g)=sup_{x\in X}|f(x)-g(x)|$. Show that ...
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38 views

Bounded linear functionals over smooth maps of a compact interval

I have two questions regarding the topological dual of the space $E = \mathcal{C}^\infty([0; 1])$ of infinitely continuously differentiable functions over the closed interval $[0; 1]$ equipped with ...
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1answer
31 views

Net converging weak-*, implies uniform bound?

Let $E$ be a complex Banach space. A consequence of the uniform boundedness principle is the following. If $(\lambda_n)_{n\geq 1}$, $\lambda$, are elements of $E^*$ such that $$ \lambda_n(x) ...
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36 views

Choice of a dense subset of a separable Banach space

I recently came across the following statement, and still can't prove it: Statement: Suppose $X$ is a separable,closed subspace of $L^1(G)$, where $G$ is a locally compact group. Since $X$ is ...
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73 views

Sequences and reflexivity

Assume X to be a real reflexive Banach space. Why are sequential topological notions topological notions ? (relatively to the weak topology on X and the weak star topology on X*) For ex : sequentially ...
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59 views

Show that $c_0$ is a Banach space with the norm $\rVert \cdot \lVert_\infty$

Let $ c_0 = \{ x = \{x_n\}_{n \in \mathbb N} \in l^\infty : lim_{n \rightarrow \infty} x_n = 0\}$. Show that $c_0$ is a Banach space with the norm $\rVert \cdot \lVert_\infty$ I am capable of showing ...
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61 views

Prove/disprove space is complete with metric defined by an integral (triangle inequality still missing in metric part)

I have a two part question: I need to show that $d(f,g)=\int_{-1}^1\! |f(x)-g(x)| \, \mathrm{d}x$ is a metric in $C((-1,1),\mathbb{R)}$ and furthermore prove/disprove that the space ...
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52 views

Does (L) sets in a dual Banach space X* are weak* precompact? weak* sequentially precompact?

Let $X$ be a Banach space. A subset $B$ of the dual $X$ is said to be $(L)$ set if any weakly null sequence $(x_n)\in X$ converges uniformly to zero on $B$. It is well Known in the theory that ...
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27 views

Is $U\times\{y\}$ ever open in $X\times Y$, here $X$ and $Y$ are Banach spaces?

Let $U$ be an open set in a Banach space $X$, let $Y$ be another Banach space, and let $y\in Y$. Is it ever the case that \begin{align*} U\times\{y\} \end{align*} is an open set in $X\times Y$?
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10 views

Norms detecting singularities for functions with spikes

For a fixed $0<\alpha<1$ consider a segment $I=[0,1]$ and a subspace $B$ of $L_1(I)$ of functions $\phi$ than can be represented as $$ \phi = \phi_{reg} + \sum_{i=0}^\infty \frac{\phi_k}{ |x - ...
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27 views

is $\langle\lim_{n\to \infty}u_n,g\rangle = \lim_{n\to\infty} \langle u,g\rangle $ valid for bounded linear operators?

Suppose M is any linear manifold in H. H is a hilbert space. Define the orthogonal complement of M to be $$M' =\{f \in H | \langle f,g\rangle= 0 ,\forall g\in M\}.$$ To see that M' is a closed ...
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24 views

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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1answer
36 views

Are dual spaces with unconditional bases 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 converse to this statement true? If a Banach space is a ...
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19 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 ...
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1answer
38 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. ...
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39 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 ...
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51 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) = ...
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42 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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87 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
57 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 ...
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3answers
86 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 ...
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23 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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40 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 ...
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1answer
92 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 ...
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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 ...
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1answer
75 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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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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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?
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71 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 ...
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1answer
17 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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74 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 ...
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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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2answers
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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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47 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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1answer
121 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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65 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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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
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1answer
79 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
42 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
87 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) $ ...
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1answer
22 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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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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29 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 ...
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35 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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28 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
28 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 $ ...
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1answer
39 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 ...