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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28 views
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
$$||f|| = \sup_{a\leq x\leq ...
2
votes
1answer
35 views
A Banach space with multiple preduals
Is it possible to have two (separable) Banach spaces, $X$ and $Y$, that are not isometrically isomorphic, and yet their dual spaces $X^*$ and $Y^*$ are isometrically isomorphic?
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votes
1answer
23 views
existence of a weakly cauchy sequence if the dual space is separable [on hold]
Let X be a normed space such that $X^*$ is separable. Given any sequence $(x_n)\in X$ then there exist a subsequence weakly cauchy
1
vote
2answers
19 views
Understanding the Frechet derivative
What is the relationship between the normal 'high school' concept of a derivative and a Frechet derivative? According to wikipedia the Frechet "extends the idea of the derivative from real-valued ...
4
votes
1answer
32 views
A detail in the proof of Banach-Steinhaus theorem that I don't understand
I am studying functional analysis and I have seen the Banach-Steinhaus
theorem.
For starters, the motivation given was the question about when $\{T_{\alpha}\}_{\alpha\in A}$
are bounded by $M$ (here ...
0
votes
0answers
25 views
Show that $(D_t t)$ is an isomorphism.
Let $B$ be a Banach space, $\epsilon > 0$, and
$$C_1^0 ([-\epsilon,\epsilon],B) = \{u \in C^0([-\epsilon,\epsilon],B) : tu \in C^1([-\epsilon,\epsilon],B)\}.$$
Denote $\partial/\partial t$ by ...
0
votes
1answer
33 views
Isometry from Banach Space to a Normed linear space maps
Let $X$ be a Banach Space and $Y$ be a normed linear space. Show that if $T$ is an isometry then $T(X)$ is closed in $Y$.
Let me have some idea to solve this. Thank you for your help.
3
votes
1answer
33 views
Projective limit of Banach spaces
Let $(X_s)_{s \in (0,s_1)}$ be an increasing sequence of Banach spaces with the property that if $0<s<r<s_1$, then
$$ \|u\|_{X_s} \leq \|u\|_{X_r}. $$
We define
$$\tilde{X}_s = ...
3
votes
1answer
25 views
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 $x$ ...
0
votes
1answer
20 views
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
2
votes
1answer
29 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.
2
votes
0answers
33 views
Proving norm equivalence in $W^{1-1/p,p}(\Omega)$
Define for $p\in [1,\infty)$ and $\Omega=(0,1)^N\subset\mathbb{R}^N$ $$W^{1-1/p,p}(\Omega)=\left\{u\in L^p(\Omega): \ \int_\Omega\int_\Omega\frac{|u(x)-u(y)|^p}{|x-y|^{N-1+p}}dxdy<\infty\right\}$$
...
1
vote
0answers
41 views
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 ...
2
votes
1answer
40 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 ...
1
vote
1answer
25 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 ...
0
votes
1answer
17 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 $B$.
...
3
votes
1answer
28 views
$B$ be an Infinite dimensional Banach Space and $T:B\to B$ be an continuos operator
Let $B$ be an Infinite dimensional Banach Space and $T:B\to B$ be an continuous operator such that $T(B)=B$ and $T(x)=0\Rightarrow x=0$
which of the following is correct?
$T$ maps bounded sets into ...
5
votes
1answer
73 views
Complemented Banach spaces.
Let $X$ be Banach space and $Y$ a closed subspace of $X$. Assume that there exist a closed "subset" $Z$ of $X$ with the properties:
$Z\cap Y=\{0\}$ and every $x\in X$ can be written in a unique ...
0
votes
1answer
26 views
characterization of an infinite matrix mapping and continuity
Show that an infinite matrix mapping $A=[a_{ij}]$ $:l^{\infty}\to l^{\infty}$ is continuous iff $sup_{i\in \mathbb N}$ $ \sum_{k=1}^{\infty}{|a_{ij}|}=||A||<\infty$. Give a characterization of the ...
0
votes
1answer
16 views
proving that bs is banach
Let's define $B_s$ as the of real valued sequences $(x_n)$, such that $sup_{N\in \mathbb N} |\sum_{k=0}^{N}{x_k}| $ is bounded, and make it a vector space considering the usual pointwise operations ...
0
votes
1answer
21 views
is this banach space separable?
Let consider the set $c$ of convergent sequences, and the subspace of convergent sequences to zero. They are Banach spaces over $\mathbb C, \mathbb R $ under the sup-norm (and the usual vector space ...
2
votes
2answers
40 views
Linear combinations of delta measures
Let us consider the space of Borel, regular, complex measures on the real line, endowed with the total variation norm.
Inside this space, I would like to characterize the space of all the finite ...
0
votes
1answer
24 views
Exercise in Folland on extending a closed subspace of a Banach space
Folland gives the following problem on page $159$ of his book Real Analysis:
Let $\mathcal X$ be a normed space.
If $\mathcal M$ is a closed subspace and $x\in \mathcal X \setminus \mathcal ...
2
votes
0answers
26 views
Is space of Dirac measures Banach?
Is the space of all Dirac measures on a set $\Omega$ Banach? With the total variation norm. I don't know what convergence means in this norm.. I mean how do I even think about it.
2
votes
1answer
35 views
Is there a non-reflexive Banach space which is strictly convex?
I just come up with the fact that a space being strictly convex, does not implies it is reflexive (at least I never saw a proof of it).
How can one construct a example of a non-reflexive Banach ...
2
votes
1answer
34 views
Convergence of functionals and weak convergence
I consider a Banach space $V$ with its dual $V'$.
I had a sequence of functionals $\{f_k\}_{k\in \mathbb N} \subset V'$, and I wanted to show (strong or norm) convergence of $f_k \to f \in V'$.
I ...
3
votes
1answer
28 views
Image of the tensor product of strict maps of Banach spaces
Let $f:A\to C$ and $g:B\to D$ be bounded linear maps of Banach spaces with closed image. Will $f\widehat{\otimes}g:A\widehat{\otimes}_\pi B\to C\widehat{\otimes}_\pi D$ also have closed image? What ...
3
votes
1answer
46 views
Equivalent conditions for weak and weak-$*$ convergence
Say $E$ is a normed space over the field $\mathbb K$ ($\mathbb R$ or $\mathbb C$) and $E^{*}$ its dual space. The notations for weak and weak - $*$ convergence are $x_{n} \xrightarrow{w} x$ and ...
4
votes
1answer
31 views
The Banach space $c_0$ is $C^{\infty}$-smooth.
In this paper, J. Eells defines this notion of $C^r$-smoothness for Banach spaces:
A Banach space $E$ is $C^r$-smooth, $r \geq 0$, if there exists a nontrivial (that is, nonzero) $C^r$ function ...
1
vote
0answers
36 views
Proving the $l_p$ space is complete.
I'm trying to prove $l_p$ spaces are complete.
We have an $l_p$ space $W$. Let us take a cauchy sequence. There exists $N_0\in\Bbb{N}$ such that for $m,n>N_0$, $d(x^m,x^n)<\epsilon$. This ...
