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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The sup norm on $C[0,1]$ is not equivalent to another one, induced by some inner product

Let $\mathrm{C}[0,1]$ be the space of continuous functions $[0,1]\rightarrow \mathbb{R}$ endowed with the norm $||x||_{\infty}=\mathrm{max}_{t\in [0,1]}|x(t)|$. It is easy to verify that this norm is ...
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325 views

Reflexivity of a Banach space without the James map

The reflexivity of a Banach space is usually defined as having to be enforced by a particular isometric isomorphism. Namely the map that takes each element to the evaluation, which is already an ...
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1answer
266 views

Why is the set of compact operators closed in the space of all bounded operators between Banach spaces?

Let $X$ and $Y$ be Banach space. $B(X,Y)$ is the vector space of all bounded linear maps from $X$ to $Y$. Also, $K(X,Y)$ is the set of all compact operators from $X$ to $Y$. Why is $K(X ,Y)$ ...
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155 views

Questions about the Gateaux derivative

We defined that a function is Gateaux differentiable, if all directional derivatives exist. I just wanted to check, whether I got a few things right: Now I wanted to ask, whether it is true that if ...
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1answer
164 views

Understanding the relation of weak and weak star toplogy

I'm working with Eberlein- Smulian Theorem fromm the book "Topics in Banach Space Theory". During the proof I have seen that there is used a lot the concept of weak topology and weak star topology. ...
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69 views

Banach space with cardinality bigger than $\mathfrak{c}$.

By using the infromation contained in this post, we have that the cardinality of every Banach space is equal to its dimension, which in turn, is bigger or equal to $\mathfrak{c}$. In my area of ...
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106 views

Is this space a banach space?

Hi I want to find out whether $l^1$ with the norm $||x||:=sup_n |\sum_{i=1}^{n} x_i|$ is a Banach space. In case that you think that it is a Banach space, just say: It's a Banach space(and then I will ...
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1answer
101 views

Supremum over dense subset of banach space

Let $\{x_n\}$ be a countable dense subset of a Banach space $X$. How can I show that $$\sup_{x \in X}f(x) = \sup_{n \in \mathbb{N}}f(x_n)$$ where $f$ is continuous and real-valued??
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390 views

Frechet derivative of compact operator is compact

... this seems to be a well known fact as mentioned in this and in this manuscript. However, I was not able to find a proof or to prove it by myself. So my question is: How to prove this? Any hint ...
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1answer
45 views

Proving a Space of Real Valued sequences is Banach.

Theorem: A normed vector space $(V,||\circ||)$ is a banach space if and only if for every sequence $x_n$ in $V$ with the property that $\sum ||x_n||<\infty$ we have $\sum x_n < \infty$. ...
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1answer
97 views

sequences and hyperplanes

Consider a linearly independent sequence $(x_n)$ in a separable Banach space. Can we always find a closed hyperplane $H$ and and a finite non empty subset $F$ of $\mathbb{N}$ such that $\{x_j: j\in ...
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430 views

bilinear form defined on $ X x Y \to Z$ where $X$ is banach using uniform boundedness principl

This question has a similar that was asked before, but it's not exactly equal. Please help me with this. Let $X$ be a banach space, $Y,Z$ a normed spaces, let $B: X x Y \to Z$ be a bilinear map, such ...
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1answer
147 views

on the no existence of a continuous extension of the identity function on $c_0$ to $l^{\infty}$ not equal to $I$

Let $X$ be a Banach space. A projection $P$ is a continuous map $:P:X\to X$ such that $P^2=P$ The existence of a projection it's equivalent to the decomposition of $X= M \oplus N$ where $M,N$ are ...
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1answer
103 views

Do monotone operators have positive Frechet derivatives?

If a scalar function $f\colon \mathbb R \to \mathbb R$ is monotone and differentiable, then $f'\geq 0$. Monotonicity is generalized for an operator $A\colon V \to V^*$, where $V$ is a Banach spaces ...
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1answer
262 views

characterization of weakly convergent to zero sequences on $l^p$ for $1\le p < \infty$

Let $1\le p< \infty$. Show that a sequence $t_k = ({t_{kj}})_{j=1}^{\infty}\in l^p$ converges weakly to 0 iff $||t_k||_p$ is bounded and $\lim_k t_{kj}=0$. I proved that if $t_k$ converges weakly ...
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1answer
115 views

projection from the set of convergent sequences to convergent to zero sequences.

Let $c$ be the set of all convergent real sequences $\mathbb N \to \mathbb R$ and $c_0$ be the subspace of all the sequences that converges to zero. We can consider $c$ as a Banach space under the ...
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163 views

Norms that $C([0,1])$ to be an incomplete normed space.

I searched all of norms that $C([0,1])$ to be incomplete normed space. But I found only $\|.\|_p$ (for every $1\leq p<\infty$). Are you know another norm on $C([0,1])$ that $C([0,1])$ to be ...
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2answers
222 views

Properties of the duals of $\ell^1$ and $\ell^{\infty}$

a) True or false: (i) $(\ell^{1})^* = \ell^{\infty}$ (ii) $\ell^1 \subset (\ell^\infty)^*$ (iii) $(\ell^\infty)^* \subset \ell^1$ (iv) $(\ell^1)^{**} \subset \ell^1$ b) Give the set of dual vectors ...
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194 views

If $a$ and $b$ commute in a $C^*$-algebra and $a$ is normal, then $f(a)$ and $b$ commute for any continuous $f$

I'm trying to find a way to demonstrate the following: Let $(A,*,\|\cdot\|)$ be a unital $C^*$-algebra. If $a,b\in A$ commute and $a\in A$ is normal (i.e. $a^*a=aa^*$), then for every continuous ...
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42 views

Open maping theorem. Completeness assumption are important [duplicate]

The open maping theorem between banach spaces says. Let $T:X\to Y$ be a linear,continuous and surjective map between the banach spaces $X,Y$ then $T$ is an open map. I need examples to show that the ...
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82 views

Banach spaces over complete fields with their own absolute value

Let $F\hspace{.03 in}$ be a field, and let $E\hspace{.03 in}$ be an ordered subfield of $F$. Let $\;\; |\hspace{-0.03 in}\cdot\hspace{-0.03 in}| \: : \: F \: \to \: E \;\;$ be such that for all ...
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Continuous mapping on unit ball such that $T(\Bbb x_0)=0$

got a question from a course in functional analysis. " Let $T:\{\Bbb x\in\Bbb R: ||\Bbb x||\leq 1\}\to\Bbb R^n$ be a continuous mapping. Moreover assume that $\langle T(\Bbb x),\Bbb x\rangle>0$ ...
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329 views

Showing infinite direct sum of Banach spaces with a certain norm is a Banach space

Given a family $(A_{\lambda})_{\lambda\in\Lambda}$ of Banach spaces, let $\bigoplus_{\lambda}A_{\lambda}$ be the set of all $(a_{\lambda})\in\prod_{\lambda}A_{\lambda}$ such that ...
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1answer
96 views

Weak* sequentiality

Suppose we are given a Banach space $E$ such that weak* compact subsets of $E^*$ are weak* sequentially compact (for example this happens when $E$ is separable). Does it follow that if $A$ is a subset ...
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1answer
143 views

$C_0^\infty(0,T)\cdot V$ dense in the Bochner space $L^2(0,T;V)$

Let $V$ be a Banach space and $(0,T)$ a time interval. Consider the space $C_0^\infty(0,T)$ of infinitely often differentiable functions with values in $\mathbb R$ and compact support in $(0,T)$ and ...
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81 views

Tensor products of weakly compact sets

Let $X$ and $Y$ be Banach spaces. Denote by $X\otimes_\varepsilon Y$ the injective tensor product of $X$ and $Y$. Also, let $A\subset X, B\subset Y$ be any sets. Set $A\otimes B = \{a\otimes b\colon ...
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85 views

Banach spaces and Normed linear spaces

Here's a theorem: A normed linear space X is a Banach space iff every absolutely convergent series in X is convergent. How is this possible? I need the proof.
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198 views

How to show that the dual of $(\mathbb{R}^n,\|{\cdot}\|_p)$ is $(\mathbb{R}^n,\|{\cdot}\|_q)$?

I am trying to brush up on my functional analysis and I learn some $L_p$ spaces since I was never formally intrduced to them through courses. I wanted to know if anyone could offer me a proof or give ...
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1answer
58 views

Isometries on the Banch Space M([0,1]) of regular Borel Measures

I'm trying to define an isometric isomorphism $T:M([0,1])\to M([0,1])$ that is not weak-star continuous (by $M([0,1])$ I mean the Banach space of regular Borel measures). How I can build one? One ...
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1answer
59 views

A problem of maximization in Banach spaces

Let $X$ be a Banach space and $K\subset X$ compact. Let $C(K)$ be the set of continuous function in $K$ and $\mu\in (C(K))^\star$ a non-negative measure. Assume that $f:K\to X^\star$ is a continuous ...
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Banach-Mazur distance

I am stuck upon the following problem. Consider the Banach-Mazur distance for $X$ ,$Y$ normed isomorphic vector spaces $$d(X,Y) = \inf \{ \| T \| \| T^{-1} \| : T \in GL(X,Y) \}$$ I would like to ...
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510 views

Schauder basis for a separable Banach space

It is known that if a Banach space $X$ has a Schauder basis, then $X$ is separable. On the other hand P. Enflo showed that there exist a separable Banach space without Schauder basis. If $X$ is a ...
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Norm on a Banach space

Let $\left( X, \| \cdot \|\right) $ be a Banach space over some field $\mathbb{K}$. Let $x$ be fixed in $X$ such that $\|x\| \le 1$. If $x_0$ is any point in $X$ , I need to show that there ...
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Prove that $C^1([a,b])$ with the $C^1$- norm is a Banach Space

Consider the space of continuously differentiable functions, $$C^1([a,b]) = \{f:[a,b]\rightarrow \mathbb{R}|f,f' \text{are continuous}\}$$ with the $C^1$-norm $$||f|| = \sup_{a\leq x\leq ...
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231 views

Closed-form expressions for dual norms of real normed vector spaces

Say that $V$ is a finite-dimensional real normed vector space, where for some $v \in V$ the norm is notated by $\|v\|$. Then say that $V^*$ is the dual space of linear functionals on $V$. The "dual ...
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136 views

A Banach space with multiple preduals

Is it possible to have two (separable) Banach spaces, $X$ and $Y$, that are not isometrically isomorphic, and yet their dual spaces $X^*$ and $Y^*$ are isometrically isomorphic?
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existence of a weakly cauchy sequence if the dual space is separable [closed]

Let X be a normed space such that $X^*$ is separable. Given any sequence $(x_n)\in X$ then there exist a subsequence weakly cauchy
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Understanding the Frechet derivative

What is the relationship between the normal 'high school' concept of a derivative and a Frechet derivative? According to wikipedia the Frechet "extends the idea of the derivative from real-valued ...
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1answer
283 views

A detail in the proof of Banach-Steinhaus theorem that I don't understand

I am studying functional analysis and I have seen the Banach-Steinhaus theorem. For starters, the motivation given was the question about when $\{T_{\alpha}\}_{\alpha\in A}$ are bounded by $M$ (here ...
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Show that $(D_t t)$ is an isomorphism.

Let $B$ be a Banach space, $\epsilon > 0$, and $$C_1^0 ([-\epsilon,\epsilon],B) = \{u \in C^0([-\epsilon,\epsilon],B) : tu \in C^1([-\epsilon,\epsilon],B)\}.$$ Denote $\partial/\partial t$ by ...
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Isometry from Banach Space to a Normed linear space maps

Let $X$ be a Banach Space and $Y$ be a normed linear space. Show that if $T$ is an isometry then $T(X)$ is closed in $Y$. Let me have some idea to solve this. Thank you for your help.
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Projective limit of Banach spaces

Let $(X_s)_{s \in (0,s_1)}$ be an increasing sequence of Banach spaces with the property that if $0<s<r<s_1$, then $$ \|u\|_{X_s} \leq \|u\|_{X_r}. $$ We define $$\tilde{X}_s = ...
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Banach fixed point theorem and inverse function

Let $U$ and $V$ be the open subsets in $\mathbb{R}^n$, $x\in U$ and $f:U\rightarrow V$ is a smooth function. There is an inverse function theorem which states that if the Jacobian determinant at $x$ ...
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Topology induced bycone metric

Is cone metric define atopology as same as the topology define by ametric? I have tried to prove it by theorems that joined them
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1answer
120 views

How to show this Sobolev space is a uniformly convex space?

From the book by Kufner: How do I prove this theorem? I'd like to do it using the epsilon delta definition (see http://en.wikipedia.org/wiki/Uniformly_convex_space) if possible.
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Proving norm equivalence in $W^{1-1/p,p}(\Omega)$

Define for $p\in [1,\infty)$ and $\Omega=(0,1)^N\subset\mathbb{R}^N$ $$W^{1-1/p,p}(\Omega)=\left\{u\in L^p(\Omega): \ \int_\Omega\int_\Omega\frac{|u(x)-u(y)|^p}{|x-y|^{N-1+p}}dxdy<\infty\right\}$$ ...
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Representation of subspaces as complemented subspaces

Let $X$ be any separable Banach space. The Banach-Mazur theorem states (astonishingly) that $X$ is isometric to a closed subspace of $C(\Delta)$, the space of continuous functions on the cantor set ...
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186 views

Proving that $X/M$ is a Banach space when $X$ is

I am trying to do an exercise in an introduction to functional analysis course: 1) Let $X$ be a normed space and $\{x_{n}\}_{n=1}^{\infty}\subseteq X$. Prove that $X$ is a banach space iff ...
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38 views

if $P\Rightarrow Q$ Then both are banach space?

$X,Y$ are norm linear space and $T_n$ be a sequence of bounded linear operators from $X\to Y$ consider the two statements below $P:\{\|T_n(x)\|\}$ is bounded for ever $n$ $Q:\{\|T_n\|\}$ is bounded ...
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32 views

Clarification about the space $C^0 ([-T,T],B)$

Let $B$ be a Banach space (in particular, $B$ is a function space equipped with the supremum norm). The space $C^0 ([-T,T],B)$ is the set of continuous functions on $[-T,T]$, valued in $B$. ...