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

learn more… | top users | synonyms

7
votes
2answers
1k views

Kernel of $T$ is closed iff $T$ is continuous

I know that for a Banach space $X$ and a linear functional $T:X\rightarrow\mathbb{R}$ in its dual $X'$ the following holds: \begin{align}T \text{ is continuous } \iff \text{Ker }T \text{ is ...
6
votes
1answer
339 views

Rainwater theorem, convergence of nets, initial topology

I've stumbled upon a result called Rainwater's theorem a few times, it seems to be a very useful result in connection with weak convergence in Banach spaces. Rainwater's theorem. Let $X$ be a ...
6
votes
2answers
135 views

Inclusion of $\mathbb{L}^p$ spaces, reloaded

I have a follow-up from this question. It was proved that, if $X$ is a linear subspace of $\mathbb{L}^1 (\mathbb{R})$ such that: $X$ is closed in $\mathbb{L}^1 (\mathbb{R})$; $X \subset \bigcup_{p ...
5
votes
1answer
137 views

If $P$ has marginals $P_1, P_2$, is $L^1(P_1) + L^1(P_2)$ closed in $L^1(P)$?

Suppose that $\mathbb{X}=\mathbb{X}_1\times \mathbb{X}_2$ and suppose that $ P$ is a probability measure on $\mathbb{X}$ with marginals $ P_i$ on $\mathbb{X}_i, i=1,2$, i.e., $$\int f_i(x_i)\, ...
5
votes
1answer
547 views

Proving that Tensor Product is Associative

I want to show that $X\otimes(Y\otimes Z)$ is isomorphic to $(X\otimes Y)\otimes Z$. Intuitively I think I should just choose bases $\{e_{i}\}_{i\in I}, \{f_{j}\}_{j\in J}$, and $\{g_{k}\}_{k\in K}$ ...
5
votes
2answers
213 views

Boundary of $L^1$ space

Is there any rigorous or heuristic notion of boundary of $L^1$ that is studied? I mean something loosely like the collection of functions or distributions defined by $$\left\{f\notin L^1: f_n\to ...
5
votes
2answers
336 views

Isomorphism of Banach spaces implies isomorphism of duals?

I can't make up my mind whether this question is trivial, or simply wrong, so i decided to ask, just in case someone sees a fallacy in my reasoning: Question: Suppose $V,W$ are two banach spaces, and ...
5
votes
1answer
258 views

Show sequence equicontinuous

I don't know how to prove this question: Let $X$ be a compact space, and let $(T_{n})$ be a sequence of positive linear operators on $C(X)$. Also, let $f \in C(X)$ be a strictly positive function. ...
5
votes
1answer
304 views

Is kernel a complemented subspace

Let $\mathcal{A}:X\to Y$ be continuous linear operator, $X$ and $Y$ are Banach spaces. Let $\text{Im} \mathcal{A}=Y$. Is $\ker\mathcal{A}$ a complemented subspace of $X$?
5
votes
1answer
239 views

Showing that $l^p(\mathbb{N})^* \cong l^q(\mathbb{N})$

I'm reading functional analysis in the summer, and have come to this exercise, asking to show that the two spaces $l^p(\mathbb{N})^*,l^q(\mathbb{N})$ are isomorphic, that is, by showing that every $l ...
5
votes
2answers
607 views

construction of a linear functional in $\mathcal{C}([0,1])$

Can someone help me to construct a linear functional in $\mathcal{C}([0,1])$ that does not attain its norm? Actually, I want to prove that $\mathcal{C}([0,1])$ is not reflexive Banach space. Is it ...
4
votes
2answers
776 views

Confused about which Hölder spaces are Banach

If $\Omega$ is an open set in $\mathbb{R}^n$, is the Hölder space $C^{k, \alpha}(\Omega)$ Banach? Or is it only that $C^{k, \alpha}(\overline{\Omega})$ is Banach, like with ordinary continuous ...
4
votes
1answer
274 views

Cancellation law for Minkowski sums

Let $(X,\|\cdot\|)$ be a Banach space and $A,B,C\subset X$ closed bounded non-empty convex subsets. Let $+$ denote the Minkowski symbol for addition. Does the $+$ satisfy: $$A+C\subset B+C\implies ...
4
votes
1answer
405 views

Finding the topological complement of a finite dimensional subspace

I know that for any finite dimensional subspace $F$ of a banach space $X$, there is always a closed subspace $W$ such that $X=W\oplus F$, that is, any finite dimensional subspace of a banach space is ...
4
votes
1answer
481 views

space of bounded measurable functions

Let $(\Omega, \Sigma)$ be a measurable space. Is the space of bounded measurable functions $B_b(\Sigma)$ equipped with the supremum norm a Banach space, i.e. complete?
3
votes
1answer
71 views

Is $W_0^{1,p}(\Omega)\cap L^\infty(\Omega)$ complete?

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain. Define $W=W_0^{1,p}(\Omega)\cap L^\infty(\Omega)$. Is $W$ complete with respect to the norm $\|v\|=\|v\|_{1,p}+\|v\|_\infty$? If $u_n$ is a ...
3
votes
2answers
461 views

Show that a finite-dimensional Banach space has a bijective compact operator

It is clear that if $ T: X \rightarrow X $ is a bijective compact operator, where $ X $ is a Banach space, then $ \dim(\text{Range}(T)) = \dim(X) $, which implies that $ \dim(X) $ must be $ < ...
2
votes
1answer
70 views

In a uniformly convex Banach space $x_n\stackrel{w}\to x$ and $||x_n||\to ||x||$ implies $||x_n-x||\to 0$

In a uniformly convex Banach space $$x_n\stackrel{w}\to x \ \ \text{and} \ \ ||x_n||\to ||x|| \ \ \text{implies} \ ||x_n-x||\to 0.$$ Can you help me to solve it? Thanks in advance.
2
votes
1answer
52 views

The importance of basis constant

Let $X$ be a Banach space and let $(e_n)_{n=1}^{\infty}$ be a Schauder basis for $X$. Let us denote the natural projections associated with $(e_n)_{n=1}^{\infty}$ by $(S_n)_{n=1}^{\infty}$. Then by ...
2
votes
2answers
703 views

Weak limit and strong limit

Let $X$ be a Banach space and let $x_n \overbrace{\rightarrow}^w x$ and $x_n \overbrace{\rightarrow}^s z$ can we then say that $x = z$? My try: $$\| x- z\| = \sup_{\ell \leq 1} |\ell(x-z)| = ...
2
votes
3answers
392 views

$C[0,1]$ is NOT a Banach Space w.r.t $\|\cdot\|_2$

I'm trying to find a cauchy sequence in $C[0,1]$ that converges under $\|\cdot\|_2$ to a limit which isn't continuous. Any ideas?
1
vote
2answers
45 views

Frechet differentiable implies reflexive?

Note: The question has been cross-posted (and answered) on MathOverflow here. Let $X$ be a Banach space. Is it true that if dual norm of $X^*$ is Frechet differentiable then $X$ is reflexive?
1
vote
1answer
72 views

Continued matrices-valued function

Given $d<k$. Let ${\cal M}_{d\times k}(\mathbb{R})$ denotes the set of all $d\times k$ real matrices and suppose that $H:\mathbb{R}^k\rightarrow {\cal M}_{d\times k}(\mathbb{R})$ is a continuous ...
1
vote
1answer
74 views

Is the dual of a reflexive Banach space strictly convex?

Is the dual of a reflexive Banach space strictly convex? Why? This is a question that arouse trying to understand the theory behind approximation by finite element methods.
1
vote
2answers
786 views

Hilbert Space is reflexive

A normed space $X$ is reflexive iff $X^{**}=\{g_x:x\in X\}$ where $g_x$ is bounded linear functional on $X^*$ defined by $g_x(f)=f(x)$ for any $f\in X^*$. Let $X$ be a Hilbert space, would you help ...
1
vote
1answer
223 views

To construct a counterexample of normed space

Please construct a counterexample for the following: $A$ is normed space and $M$ is a dense subspace of $A$, if there is a functional $f$ such that $f(M) = 0$, then $f=0$. Besides, if $A$ is a Banach ...
1
vote
1answer
271 views

Complementability of von Neumann algebras

Is every von Neumann algebra complemented in its bidual? It is certainly true for commutative von Neumann algebras as their spectrum is hyperstonian. Is it 1-complemented?
0
votes
1answer
76 views

every denting point and strongly exposed point is extreme point

If $X$ be a Banach space and $K$ is a subset of $X$, then I want to prove Every denting point of $K$ is extreme point Every strongly exposed point of $K$ is extreme point $K$ is the closed convex ...
0
votes
1answer
43 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 ...
9
votes
2answers
207 views

If $V \times W$ with the product norm is complete, must $V$ and $W$ be complete?

Let $V,W$ be two normed vector spaces (over a field $K$). Then their product $V \times W$ with the norm $\|(x,y)\| = \|x\|_V + \|y\|_W$ is a normed space. Using this norm it's easy to show that if ...
8
votes
1answer
194 views

Monotone convergence to a fixpoint in a Banach space

Let $\mathscr X$ be a complete separable metric space and $\mathbb B$ be the Banach space of all real-valued bounded measurable functions on $\mathscr X$. The partial order on this space is introduced ...
7
votes
1answer
271 views

separable Banach space with Banach-Mazur distances to $\ell_2^n$ bounded must be isomorphic to $\ell_2$?

If $X$ is a separable infinite-dimensional Banach space and $C\in\mathbb{R}^+$ is an upper bound for the Banach-Mazur distance $d(E,\ell_2^n)$ for all $n\in\mathbb{N}$ and all $n$-dimensional $E\leq ...
6
votes
2answers
290 views

Can a proper vector subspace of a Banach space be a countable intersection of dense open subsets?

Let $V$ be a complete normed space. Let $W$ be a proper vector subspace. Can $W$ be the intersection of a sequence of open dense subsets of $V$? If there exists an open dense proper vector subspace ...
5
votes
2answers
372 views

Totally bounded space

Suppose $M=\left \{ f\in L^1([0,1])\, |\, 0<f(x)<\frac 1{\sqrt x} \text{almost everywhere on} \, (0,1) \right\}$. Is it true or not, that $M$ is totally bounded?
5
votes
4answers
182 views

Is there a null sequence that is in no $\ell_p$ with $p<\infty$?

Is $\bigcup_{p<\infty}\ell_p=c_0$? At least one inclusion obvious: every p-summable sequence converges to zero.
5
votes
1answer
840 views

Is $p$-norm decreasing in $p$?

I could show that $\|\cdot\|_p$ is decreasing in $p$ for $p\in (0,\infty)$ in $\mathbb{R}^n$. Following are the details. Let $0<p<q$. We need to show that $\|x\|_p\ge \|x\|_q$, where $x\in ...
5
votes
2answers
787 views

Dual space of the space of finite measures

Since I am reading some stuff about weak convergence of probability measures, I started to wonder what is the dual space of the space consisting of all the finite (signed) measures (which is well ...
4
votes
1answer
81 views

What makes compact operators special?

I would like to understand why compact operators are considered so special to consider them as an extra class of operators. Over Hilbert spaces these (as far as I know) these are the ones with ...
4
votes
0answers
60 views

Equivalence of definitions of $C^k(\overline U)$

let $U$ be an open set of $\mathbb{R}^n$, that contains at least some open set. In Evans book we find the definition $$C^k(\overline U)=\{f \in C^k(U): D^\alpha f \text{ is uniformly continuous on ...
4
votes
1answer
585 views

Unit ball of a Separable Banach Spaces is metrizable

Please help me to understand why the unit ball of a separable banach space is metrizable, when it is given the induced topology from the weak topology. Specifically, my problem is I think I'd be able ...
4
votes
1answer
135 views

1-separated sequences of unit vectors in Banach spaces

Given an infinite-dimensional Banach space $X$, I would like to construct a sequence of linearly independent unit vectors such that $\|u_k-u_l\|\geqslant 1$ whenever $k\neq l$. Any ideas on how to ...
4
votes
1answer
169 views

How to prove that this metric space is complete?

Let $X=\mathrm{Conv}(\mathbf R)$, the collection of all convergent sequences in $\mathbf{R}$. Is the normed space $(X,\|\cdot\|_\infty)$ complete?
4
votes
1answer
88 views

The reflexivity of the product $L^p(I)\times L^p(I)$

I am starting to read about the Sobolev spaces $W^{1,p}(I),$ where $I$ is an open interval in $\mathbb{R}.$ In order to establish the reflexivity of $W^{1,p}(I)$ for $p\in ]1,\infty[,$ I need the ...
4
votes
1answer
290 views

Simple example of not a Banach space. Product topology.

Claim: $$R^\infty \text{ is not a Banach space when equipped with its natural product topology}$$ I need help proving this 'obvious' claim. I just got acquainted with a definition of a product ...
3
votes
1answer
181 views

Proof of equicontinuous and pointwise bounded implies compact

I tried to prove the Arzela-Ascoli theorem: Let $X$ be a compact Hausdorff space and let $C(X)$ denote the space of continuous functions $f: X \to \mathbb R$ endowed with the sup norm $\|\cdot ...
3
votes
1answer
171 views

A Banach space is reflexive if a closed subspace and its quotient space are both reflexive

Let $X$ be a Banach space. Let $Y$ be a closed subspace. Suppose that the normed spaces (in fact Banach spaces) $Y$ and $X/Y$ are both reflexive. I need to show that $X$ is reflexive. I cannot show ...
3
votes
1answer
83 views

wanted: a lucid demonstration that these $m+n$ bilinear equations have a solution

I shall first state the problem, and if it arouses any interest, I may gradually add a few notes, to provide a little context for those who, like myself, appreciate such background information. let me ...
3
votes
1answer
70 views

$L(\ell_{p})$ contains only one proper closed ideal

I am trying to solve the following problem: Show that if $1<p<\infty$ and $T:\ell_{p}\rightarrow\ell_{p}$ is not compact then there is a complemented infinite dimensional subspace $E$ of ...
3
votes
1answer
65 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
671 views

Weakly compact implies bounded in norm [duplicate]

The weak topology on a normed vector space $X$ is the weakest topology making every bounded linear functionals $x^*\in X^*$ continuous. If a subset $C$ of $X$ is compact for the weak topology, then ...