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

1
vote
2answers
24 views

Using a Quotient map to induce a well-defined map

Let $X, Y, Z$ be Banach spaces, and let $q:X\to Y$ be a quotient map. Given a bounded linear operator $T:X\to Z$, does there exist some criteria on $T$ for determining when $q$ can be used to induce ...
0
votes
0answers
23 views

Verification: Closed Set Expands to Fill Space, but Contains No Open Ball $B_\epsilon(0) $?

I have the proof that $C$ closed, convex, symmetric in Banach space $X$ and $\cup_{n \in N \setminus 0} n.C= X $ then $B_\epsilon(0) \subset C$ for some $\epsilon > 0$. I also have the proof for $...
1
vote
1answer
24 views

Show space of C1 functions on (0,1) is a Banach lattice

I'm working on the beginning of a book on Banach lattices, and it wants me to show that $C^1(0,1)$ is a Banach Lattice, with the norm $\|f\|=\|f'\|_\infty+|f(0)|$ and the order $f\le g$ if $f(0)\le f(...
0
votes
1answer
20 views

Proof Verification: C closed, convex, symmetric in Banach space X and $\cup_{n \in N \setminus 0} n.C= X$ then $B_\epsilon(0) \in C $.

I have an outline of the proof of this which I've expanded (correctly or otherwise) below, I'd appreciate feedback on it. (I think that C has to be closed in order to assert that $\cup_{n \in N \...
2
votes
1answer
47 views

Is $C[a,b]$ isomorphic to $C([a,b]\cup[c,d])$

It is very simple to prove that $C[a,b]$ is isomorphic to subspace of $C([a,b]\cup[c,d])$ and vice versa $C([a,b]\cup[c,d])$ is isomorphic to subspace of $C[a,b]$. Is it true for $\bigl(C[a,b], C([a,...
1
vote
2answers
34 views

$Im(K)\subset Y$ closed, infinite-dimensional: $K:X\to Y$ is not a compact operator

Proposition: If $K:X\to Y$ is a bounded linear operator between two Banach spaces $X$ and $Y$ such that $\operatorname{Im}(K)\subset Y$ is an infinite-dimensional closed subspace, then $K$ is NOT ...
1
vote
1answer
49 views

Space of $f(0)=f(1)$: Is Hilbert space?

Let $S$ be space consisting of collection of square integrable continuous functions $f:[0,1]\rightarrow\mathbb{R}$ with the constraint $f(0)=f(1)$. So $S$ is an inner product space with the inner ...
0
votes
0answers
18 views

How to prove $\limsup\limits_{n\to\infty}\rho_k(x_n+x)=\limsup\limits_{n\to\infty}\rho_k(x_n)+\rho(x)?$ on $\ell_1$

Let $p(.)$ be an equivalent norm to the usual norm on $\ell_1$ such that $$\limsup\limits_{n\to\infty} p(x_n+x)=\limsup\limits_{n\to\infty}p(x_n)+p(x)$$ for every $w^*-$null sequence $(x_n)$ and for ...
2
votes
0answers
28 views

About the Banach algebra $\ell^{\infty} (K( \ell^{1}))$

If $P_{n}: \ell^{1} \rightarrow \ell^{1}$ is the projection onto the first n coordinates, then it's well known that $ P_{n} K(\ell^{1}) P_{n}$ is isomorphic to $B(\ell^{1}_{n})$, the Banach space of ...
1
vote
2answers
52 views

Prove or disprove: $\{t^{2k}\}_{k=0}^{\infty}$ complete in $L_2[-1,3]$

Is $\{t^{2k}\}_{k=0}^{\infty}$ not complete in $L_2[-1,3]$?(Here, completeness of a system is equivalent to the density of its span) Obviously many polynomials in the domain will be irreleant, but I ...
2
votes
0answers
35 views

When can I say that $\overline{A} \subset B$ if I know that $A \subset B$?

my question is as stated in the title: When can I say that $\overline{A} \subset B$ if $A \subset B$? Here $A,B$ are normed spaces and the closure of A is taken with respect to the norm of B. Can I ...
2
votes
1answer
37 views

Construct a sequence in Banach Space

Prove the equivalence between: $\forall x \in B_E = \{y \in E:\|y\| \leq 1 \}$ $\exists (x_n) \subset E$ such that $\|x_n\|=1$ and $x_n \rightarrow_w x$ (weak convergence). $\exists (x_n) \subset E$ ...
2
votes
1answer
18 views

Any metric space can be isometrically embed in some Banach space? [duplicate]

I have just read the question of the title in an article from Kirchheim. I didn't know this result, does any one know where I can find a proof of it?
2
votes
0answers
26 views

A Banach space $X$ is reflexive iff $X^*$ is reflexive [duplicate]

I have already shown that if $X$ is reflexive then $X^*$ is reflexive, but I need some help on the other direction. The canonical mapping is defined by $$ J : X \to X^{**}, \ J(x) (f) = f(x)$$ For ...
0
votes
1answer
40 views

Projections in the lp direct sum $E=(\bigoplus_{n=1}^\infty\ell_1^n)_p$.

Fix $1<p<\infty$ and define the (reflexive) Banach space \begin{equation*}E=\left(\bigoplus_{n=1}^\infty\ell_1^n\right)_{\ell_p}.\end{equation*} Let $Y$ denote the closed subspace of $E$ ...
1
vote
0answers
33 views

Basis equivalent to the unit vector basis of $(\oplus_{n=0}^\infty\ell_\infty^{2^n})_p$

Definitions and notation. Fix $1<p<\infty$, and define the (reflexive) Banach space \begin{equation*}E=\left(\bigoplus_{n=1}^\infty\ell_\infty^{2^{n-1}}\right)_{\ell_p}.\end{equation*} It has a ...
0
votes
0answers
21 views

Image of unit ball precompact implies bounded

I need to prove that if the image of the unit ball $X_1$ of a Banach space $X$ along an operator $A:X\rightarrow Y$ is precompact, $A$ is bounded. Is my solution correct? $\|A\|=\sup_{\|x\|=1}\|Ax\|...
0
votes
0answers
40 views

Implicit function theorem for Banach spaces

I was wondering if someone could give a bit of broad advice regarding working with Implicit Function Theorem (IFT) and, I guess, the Catastrophe theory. This is something completely new to me. ...
2
votes
1answer
69 views

Determining whether equality $ \|T v\| = \|T\| \cdot \|v\|$ is possible

As an exercise, I'm supposed to determine whether for the operators $A_\lambda:g(t)\mapsto \sqrt\lambda g(\lambda t)$ on $C[0,1],\lambda\in (0,1)$ and $T_f:g\mapsto g(t)f(t)$ on $L^2$ it is possible ...
0
votes
0answers
29 views

What's the difference between the topology defined by a seminorm and the topology defined by the norm it induces?

I was just wondering whether there's some big difference between the topology generated by a seminorm and the norm it induces. For instance, Suppose $X$ is normed and $A$ is a subspace. $X/A$ is semi-...
2
votes
1answer
43 views

If $\|e_1+e_2+\cdots+e_n\|\leq C$ for all $n$ then $(e_i)_{i=1}^n$ is uniformly equivalent to the basis of $\ell_\infty^n$?

Conjecture. Let $n\in\mathbb{N}$, and let $(e_i)_{i=1}^n$ be a normalized unconditional basis for an $n$-dimensional Banach space $E_n$ with Schauder basis constant $K\in[1,\infty)$. If $\|e_1+e_2+\...
1
vote
1answer
16 views

Is a reflexive space necessarily an L-embedded space?

I am reading some paper about L-embedded space. For the definition of L-embedded space, see http://www.sciencedirect.com/science/article/pii/S0022247X02001075. Let $Y$ be a Banach space and $P$ a ...
1
vote
1answer
25 views

Question about the proof $X'$ reflexive $\Rightarrow X$ reflexive.

I have a doubt in the proof I have been given of the fact: For a Banach space $X$, if $X'$ is reflexive then $X$ is reflexive. This is proven by showing first theorem 1 and theorem 2, which I quote ...
2
votes
2answers
46 views

Existence of a function that fulfills an equation

I am just revising for my exams and came across this question: Show that in the Banach-space of functions that are continuous in the interval $[-1,1]$, together with the supremum-norm, there is ...
6
votes
1answer
53 views

If B(X) is isomorphic to B(Y), does that mean X is isomorphic to Y (for X and Y Banach spaces)?

Let $X$ and $Y$ be Banach spaces such that $\mathcal{B}(X)$ is linearly isomorphic to $\mathcal{B}(Y)$ (where $\mathcal{B}(\cdot)$ denotes the algebra of bounded linear operators). Must it always be ...
2
votes
1answer
38 views

Second adjoint of the canonical embedding

Suppose that $X$ is a Banach space. Denote by $\kappa_X$ the canonical embedding of $X$ into $X^{**}$. Do we always have $$(\kappa_X)^{**} = \kappa_{X^{**}}? $$
1
vote
2answers
56 views

A corollary of the Hahn-Banach theorem

Let $Z$ be a subspace of normed linear space $X$ and that $y$ is an element of $X$ whose distance from $Z$ is $d$. Then there exists a $\Lambda \in X^* $ (the dual space of $X$) so that $\| \Lambda\| \...
1
vote
1answer
43 views

Showing basic properties of Riemann integral

I'm (recreationally) trying to see to what extent Riemann integration can be extended to functions of the form $f : [a, b] \to B$ where $[a, b] \subseteq \mathbb{R}$ is some compact interval and $(B, \...
3
votes
1answer
41 views

Is $AC[a,b]$ closed in $(BV[a,b],TV)$?

Consider $BV[a,b]$ the space of all bounded variation functions on a real interval $[a,b]$, endowed with the total variation norm $TV$. $AC[a,b]$, the space of absolutely continuous functions, is a ...
0
votes
0answers
47 views

Total variation on BV functions: “Banach seminorm”?

Suppose I consider the space $BV[a,b]$ of all bounded variation functions on $[a,b]$ a real interval. I endow it with $\|f\|=TV(f)$ the total variation norm. Do I get a Banach space? How can I prove ...
0
votes
1answer
32 views

How to use the completeness here

Suppose that $G\subset \mathbb{C}$ is an open subset and that $X$ is a Banach space. Fix $z_0 \in G$ and let $V:=\{(h,k) \in \mathbb{C}^*\times\mathbb{C}^* : z_0+h \in G\ ; z_0 + k \in G\}$. Let $f: G ...
2
votes
1answer
53 views

Banach-Space-Valued Analytic Functions

This is Chapter VII, $\S$3, exercise 4, from Conway's book: A Course in Functional Analysis: Let $X$ be a Banach space and $G\subset \mathbb{C}$ an open subset. We say that $f: G \to X$ is analytic ...
0
votes
1answer
14 views

Do null sequences in Banach space have summable subsequences?

One of the very nice features of null scalar sequences is the fact that they admit summable subsequences. Is the same true in Banach spaces? That is, if $(x_n)_{n=1}^\infty$ is a sequence in a Banach ...
1
vote
0answers
16 views

Fixed points and banach spaces [closed]

Let $B$ be a closed ball centered at $0$ in a Banach space $E$ and $F:B\to E$ be a contractive map such that $F(x)=-F(-x)$ for every $x\in\partial B$. Show that $F$ has a fixed point. Any ideas?
2
votes
3answers
75 views

Uncountable sets in metric spaces

Suppose we work in a separable, complete metric space $X$. Let $Z$ be an uncountable subset of $X$, must there exist $x_0\in Z$ and a sequence $(x_n)_{n=1}^\infty$ of elements in $Z$ different from $...
1
vote
1answer
34 views

Summation functional on a Hamel basis

Let $X$ be an infinite-dimensional Banach space. Is it possible to choose a Hamel basis $B$ of $X$ such that the linear functional defined by $f(b)=1$ ($b\in B$) was continuous?
0
votes
1answer
23 views

Strongly measurable implies Borel measurable in separable space

Let $(M,\mu)$ be a measure space, $X$ be a Banach space, $f: M \to X$ be a function. $f$ is said to be strongly measurable if there is a sequence of simple functions $\{f_n\}\to f$ pointwisely a.e.. $...
2
votes
2answers
29 views

About a partial converse to the Banach-Steinhaus Theorem

I've been reading the GTM text Topics in Banach Space Theory by Albaic and Kalton. In the appendix, it states the following partial converse to the Banach-Steinhaus theorem: Let $\{S_n\}$ be a ...
1
vote
0answers
25 views

Something is wrong with this argument (Lorentz and Rosenthal-Woo sequence spaces)

Fix once and for all $1<p<\infty$. Throughout, $w=(w_n)_{n=1}^\infty$ will denote a sequence of positive real weights satisfying \begin{equation}1=w_1\geq w_2\geq w_3\geq\cdots>0\;\;\;\text{ ...
1
vote
1answer
41 views

Norm of a unital homomorphism

Let $\mathcal{A}_1$ and $\mathcal{A}_2$ be two unital $C^*$-algebras and $\varphi : \mathcal{A}_1 \to \mathcal{A}_2$ a unital $*$-homomorphism, i.e. a linear map such that $\varphi(xy)=\varphi(x)\...
0
votes
0answers
22 views

Showing that $\log (\log 1+\frac{1}{|x|})$ belongs to $W^{1,p}(\Omega)$ for $p \geq 2$.

I want to show that the function $f$ belongs to $W^{1,p}(\mathbb{R}^n)$ for $p \geq 2$, where $f$ is defined as $$ f(x)=\log\left(\log \left(1+\frac{1}{|x|}\right) \right)$$ Note: This is an example ...
1
vote
0answers
38 views

Bounding $L_p$ norms on a convergent $L_1$ sequence

I've encountered a prelim problem on $L_p$ spaces that I'm pretty stuck on. Suppose $1 < p < \infty$ and $f_n \in L_1([0,1]) \cap L_p([0,1])$, with $||f_n||_p$ bounded above by some constant $M$...
0
votes
1answer
17 views

Relationship between spectral rays commuting

Show sing binomial Newton that if $S$ and $T$ are continuous operators on Banach space $B$ such that $ST = TS$ then $$r_\sigma(T + S) \leq r_\sigma(T) + r_\sigma(S)$$ where $r_\sigma$ is spectral ...
0
votes
1answer
26 views

Showing that if $E^*$ is reflexive, then $E$ is reflexive , for a Banach space $E$. [duplicate]

Let $E$ be a banach space. I have already shown that if $E$ is reflexive then $E^{*}$ is reflexive. Now I want to show that if the dual space $E^{*}$ is reflexive, then $E$ is reflexive. If $E^...
1
vote
1answer
14 views

Putting a Lower Bound On The Power Type Of Modulus Of Uniform Convexity

Take a Banach space $(X,\|\cdot\|_X)$ and define, on the interval $[0,2]$, the modulus of uniform convexity to be $\delta_X(\epsilon) = \inf\{1-\frac{\|x+y\|}{2}:\|x\|=\|y\|=1, \|x-y\|=\epsilon\}$. $...
0
votes
1answer
89 views

Continuous injection and density in $l_p$ spaces

If $r \le s$ then $l_r$$\subseteq$ $l_s$ . How can I prove there is a continuous injection $l_r$ $\hookrightarrow$ $l_s$? The suggestion was to use the fact that $\Vert$x$\Vert$$_r$ $\le$ $\Vert$x$\...
3
votes
0answers
41 views

What does well-isomorphic mean?

What I'm currently reading discusses the notion of spaces being well-isomorphic, specifically a Banach space containing well-isomorphic copies of $\ell_1^n$ for every $n\in\mathbb{N}$. The author ...
1
vote
2answers
90 views

Restriction of operators on $l_\infty$ to $c_0$

Given $\epsilon>0$, can we always find a non-compact operator $T:l_\infty\to l_\infty$ of norm larger than $1$ such that the restriction of $T$ to $c_0$ is compact and has norm smaller than $\...
1
vote
1answer
43 views

$B$ be an uncountable dimensional real Banach space and $T:B \to B$ be a surjective isometry such that $T(0)=0$ , then is $T$ linear?

Let $B$ be a real Banach space and $T:B \to B$ be a surjective isometry such that $T(0)=0$ , then is $T$ linear ? I know that it is true if $B$ is a Hilbert space ( we only need an inner product ...
0
votes
1answer
36 views

In $(C[0,1],||.||_{\infty})$ , is the set $\{p(x)\in C[0,1]$, $p(x)$ is a polynomial $:\int_0^1 p(x)dx=1\}$ totally bounded?

Consider the metric space $(C[0,1],||.||_{\infty})$ , in this space , is the set $\{p(x)\in C[0,1]$, $p(x)$ is a polynomial $:\int_0^1 p(x)dx=1\}$ totally bounded ? Please help , Thanks in advance