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

4
votes
1answer
38 views

Differentiability of an action of the group of invertible elements of a $C^{*}$-algebra $\mathcal{A}$ on the dual of $\mathcal{A}$

I am studying the actions of Banach-Lie groups on Banach manifolds, and I am not able to concretely evaluate the differentiability properties of a specific action. Let $\mathcal{A}$ be a unital ...
-1
votes
1answer
54 views

The space $C[a,b]$ with $\|f\|:=\int_a^b |f(x)|d(x)$ is a normed space but not a Banach space

Let $-\infty\lt a\lt b \lt \infty$. Let $C[a,b]$ is space of continuous functions and a function $\|.\|:C[a,b]\to R$ is given by $\|f\|:=\int_a^b |f(x)|d(x)$ a). Show that $\|\cdot\|$ is a norm in ...
0
votes
0answers
19 views

Extreme positive functionals on $\ell^\infty$

Let $\phi$ be a positive functional on $\ell^\infty$ such that $\phi((1,1,1,\dots))=1$ which cannot be written as a non-trivial convex combination of such functionals. Is $\phi$ necessarily ...
1
vote
0answers
31 views

Do linear transformations of convergent sequences converge?

Let $T \in \mathcal{L}(E,F)$ where $E, F$ are two normed vector spaces (not necessarily finite or complete (Banach)). Is it true that if $x_{n} \rightarrow x$ in $E$, is it true that $Tx_{n} ...
2
votes
1answer
48 views

$\|L(v)\| \leq \|L\|\cdot\|v\|$ on Banach spaces

Let $A,N$ be Banach spaces and let $L: A \rightarrow N$ be a linear transformation. If $L$ is continuous (which is guaranteed on finite dimensional spaces), the set $ \{ M \geq 0 \ : \|L(v)\| \leq ...
1
vote
1answer
68 views

The space of continuous functions from $\mathbb{R}$ to $\mathbb{R}$ [closed]

Is the space of continuous function from the set of the number $\mathbb{R}$ to $\mathbb{R}$ (usually denoted by $C(\mathbb{R},\mathbb{R})$ is a Banach space? With the norm $\mathop {\sup }\limits_{x ...
0
votes
1answer
26 views

Pseudo-resolvent function

Let $\emptyset \neq D$ a open set in $\mathbb{C}$ and $J: D \to B(E)$ a continuos function such that $J(\lambda) - J(\mu) = (\mu - \lambda)J(\lambda) J(\mu)$ where $E$ is Banach space. We must show ...
2
votes
1answer
36 views

Show that the closure of $C_c(X)$ is $C_0(X)$.

Let $(X,T)$ be a topological Hausdorff space. By $C_b(X)$ denote the continuous bounded function $f\colon X\to\mathbb{R}$, by $C_c(X)$ the continuous functions $f\colon X\to\mathbb{R}$ which have ...
1
vote
1answer
46 views

Show there is a continuous isomorphism $l_{\infty}\rightarrow \left(l_1\right)^* $

Let $\left(l_1\right)^*$ be the dual space to $l_1$. Each $f \in \left(l_1\right)^*$ is a continuous linear functional over $l_1$. There is constant $C \in \Bbb R$ such that $|f(x)|\le C|x|_1, \forall ...
1
vote
1answer
45 views

Show, that $c$ and $c_0$ is a Banach space

Let $c=\lbrace x=\lbrace x_n\rbrace ,n\in \mathbb{N}: \exists \, \text{lim}_{n\to \infty}x_n\rbrace$ and $c_0=\lbrace x=\lbrace x_n\rbrace ,n\in \mathbb{N}: \text{lim}_{n\to \infty}x_n=0\rbrace$. I ...
3
votes
1answer
61 views

Are the weak* and the sequential weak* closures the same?

I have a question that might be easy for the experts. Let $E$ be a Banach space (non-separable) and let $E'$ be its dual space. Suppose that $X\subset E'$ and assume that $X$ is separable with ...
1
vote
0answers
17 views

Proving Banach space is finite in Baire space [duplicate]

Let $B$ be a Banach space where the dimension of the underlying vector space is countable. using the Baire category Theorem, prove that the dimension of the underlying vector space is, in fact, ...
0
votes
0answers
22 views

On bounded bijective linear maps

If $f$ is a bounded bijective linear map from a Banach space $E$ to $E$. How can one prove that if $(f(x_n))$ converges to $0$ then $(x_n)$ also converges to $0$ without using the open map theorem.
0
votes
1answer
40 views

$\{x_n\}$ a sequence in NLS $X$ s.t. $\sum f(x_n)$ converges for all $f \in X^*$ , is the function $f \in X^* \to \sum f(x_n)$ continuous ?

Let $X$ be a NLS , $X^*$ be the set of all bounded real valued functions on $X$ ( i.e. the topological dual of $X$) , let $\{x_n\}$ be a sequence in $X$ such that $\sum_{n=1}^{\infty} f(x_n)$ ...
0
votes
1answer
109 views

Goldstine theorem and weak topology on $E$ induced by weak* topology on $E''$

In the book where I'm studying there this exercise: "(Goldstine's theorem). Recall that $E \subset E''$, or, more precisely, $J_E(E)$ is a subspace of $E''$ where $J_E :E \longrightarrow E''$ is ...
2
votes
1answer
50 views

$X$ be Banach , $T:X \to \mathcal l ^{\infty}$ be linear , $(Tx)_n$ the $n$-th term of $T(x)$;$f_n(x)=(Tx)_n$ ; if each $f_n$ is bdd then so is $T$?

Let $X$ be a Banach space , $T:X \to \mathcal l ^{\infty}$ be a linear transformation , for each $x\in X$ and each $n \in \mathbb N$ , $(Tx)_n$ be the $n$-th term of $T(x)$ and for each $n \in ...
2
votes
1answer
19 views

Proof verification in a functional analysis problem

I am new to Functional Analysis .Please review the following proof: Let $X$ be a Banach space. Let $T:X\to X$ be a invertible linear operator and $M>0$ be such that $\|T^{-k}\|<M$ for all ...
0
votes
0answers
15 views

Adjoint lattice homomorphic if surjective

Is the adjoint of a linear operator $T:X\to Y$ between Banach lattices, always lattice homomorphic if $T$ is surjective? This is my proof but I really doubt this is true: $\forall a,b\in Y'$ and ...
1
vote
1answer
32 views

Continuity on $L_p$ spaces

Consider a nonlinear and continuos function $f:\mathbb{R} \rightarrow \mathbb{R}$ and we define the functional \begin{equation} F(u) = \int_{[0,1]^2} f(u(x,y)) dxdy \end{equation} where $u$ is an ...
2
votes
1answer
21 views

Are all adjoints lattice homomorphisms?

Obviously something must be wrong in the following reasoning proving that any linear operator $T:X\to Y$ between Banach lattices has a lattice homomorphic adjoint: $\forall a,b\in E':$ $$T'(a\wedge ...
1
vote
0answers
51 views

Application of Gronwall Inequality

Let $T>0$ and $f\in C(\mathbb R, L^{2}(\mathbb R))$ with the following property: Put $g(t):= \sup\limits_{0\leq \tau\leq t} \|f(\tau)\|_{X},$ where $X \subset L^{2}$ and $X$ is a Banach Space. ...
1
vote
1answer
39 views

Sequence of dense sets in Banach spaces

Let $A_0 \supset A_1 \supset A_2 \supset \cdots$ - sequence of embedded Banach spaces and $B_0 \supset B_1 \supset B_2 \supset \cdots$ - suquence of linear spaces such that $B_i$ dense in $A_i$, it is ...
1
vote
1answer
32 views

Quotients of $L_1$

I know the rather standard fact in Banach space theory that every separable Banach space is a quotient of $\ell_1$. Is it true that every (possibly non-separable) Banach space is a quotient of some ...
1
vote
1answer
28 views

Subspaces of quotients of $L^p$ spaces

Is the collection of subspaces of quotients of $L^p$ spaces considered to be a large class of Banach spaces?
2
votes
0answers
45 views

Weak-* bounded, closed convex set is compact?

Suppose $E$ is a Banach space, and $K\subseteq E^*$ is convex, and is closed and bounded with respect to weak-* topology. Is it true that $K$ is compact? If $E$ is reflexive, then this is the case, ...
7
votes
1answer
255 views

Holomorphy of a function with values in a Hilbert space

Denote by $\mathbb C^\infty $ the Hilbert space $\ell^2(\mathbb C)$. Fix $1\leq N,M \leq \infty$, and let $U$ be an open subset of $\mathbb C^N $. Following Mujica's book "complex analysis in Banach ...
0
votes
0answers
18 views

Unable to find referenced theorem

I've am reading the article "Finitely summable Fredholm modules over higher rank groups and lattices" http://arxiv.org/abs/0806.2759 . Theorem 4.3 here refers to the article Property (T) and rigidity ...
1
vote
1answer
30 views

Locally Lipschitz continuity of the duality map

Let $E$ be a Banach space and let $F(y)$ denote the duality map: $$F(y)=\{y^*\in E^*/ \langle y^*,y\rangle =\|y\|^2=\|y^*\|^2\}$$ where $E^*$ is the dual space of $E$. Are there any sufficient ...
2
votes
1answer
31 views

The limit of a singular matrix?

Now to show the set of invertible $n \times n$ matrices are an open set in the set of all $n \times n$ matrices one can show the set of singular matrices are closed in the set of all $n \times n$ ...
1
vote
1answer
39 views

Real Analysis, Folland problem 5.3.37 Application of the Uniform Boundedness Principle

The Uniform Boundedness Principle - Suppose that $\mathscr{X}$ and $\mathscr{Y}$ are normed vector spaces and $\mathcal{A}$ is a subset of $L(\mathscr{X},\mathscr{Y})$. a.) If ...
4
votes
2answers
57 views

Does every element of the weak-star closure of a set belong to the weak-star closure of a bounded subset?

I feel like this must be a monumentally stupid question. Say $X$ is a Banach space, $S\subset X^*$, and $x^*$ is in the weak* closure of $S$. Must $x^*$ lie in the weak* closure of some norm-bounded ...
1
vote
1answer
61 views

The unit sphere is not bijective

Let $S^n = \{ x \in \mathbb R^n : ||x|| = 1 \}$ be the unit sphere, then there exists no bijection between $S^n$ and an open subset of any Banach space. How to show that? I see that $S^n$ could not ...
0
votes
1answer
52 views

Bounded and surjective map from counting $L^1$ to separable Banach space.

Let $\mathscr{X}$ be a separable Banach space and let $\mu$ be counting measure on $\mathbb{N}$. Suppose that $\{x_n\}_{1}^{\infty}$ is a countable dense subset of the unit ball of $\mathscr{X}$, ...
0
votes
1answer
53 views

Null space isomorphic to range iff closed

Problem 5.3.35 from Folland: Let $\mathscr{X}$ and $\mathscr{Y}$ be Banach spaces, $T\in L(\mathscr{X},\mathscr{Y})$, $\mathscr{N} =\{x: Tx = 0\}$, and $\mathscr{M} = range(T)$. Then ...
4
votes
0answers
46 views

Submultiplicative Hilbert space norm on $B(H)$

Let $H$ be a complex Hilbert space and let $B(H)$ denote the space of bounded linear operators $H \to H$ equipped with operator norm: $$ \lVert T \rVert = \sup\big\{ \lVert Tx \rVert \: : \: \lVert x ...
3
votes
0answers
55 views

Does the Closed Graph Theorem follow from Banach-Steinhaus?

Q: Is there a simple (but perhaps tricky or clever) proof of the Closed Graph Theorem (or the Open Mapping Theorem, or the result I call the Automatic Inverses Theorem below) from the Banach-Steinhaus ...
1
vote
0answers
11 views

Nuclear Frechet space as inductive limit

Can a nuclear Frechet space also be defined as an countable inductive system of Banach spaces with nuclear maps?
1
vote
0answers
15 views

Existence of Banach space in which nuclear space embeds densely

If $N$ is a nuclear space, does there exist a Banach space $X$, s.t. $N$ embeds densely in $X$?
1
vote
0answers
38 views

On Banach space , is every linear bounded projection map an open map?

Let $X$ be a Banach space and $P \in \mathcal B(X)$ be a projection ( i.e. $P^2=P$ ) . Is it true that $P$ is an open map in the sense that for every open set $U$ in $X$ , $P(U)$ is open in $P(X)$ ? ...
1
vote
0answers
10 views

Closed sublattice generated by countable set

Apparently the closed sublattice generated by a countable subset of a Banach lattice is separable. I am trying proof this, but I am stuck, does anybody have an idea? Trying to prove the above, I ...
2
votes
1answer
35 views

If $\{x_n\}_{n=1}^\infty$ is a basis for $X$ is $\{x_1\}\cup\{x_n-x_{n-1}\}_{n=2}^\infty$ also a basis for $X$?

Conjecture 1. Let $(x_n)_{n=1}^\infty$ be a (Schauder) basis for a Banach space $X$. Set $y_1=x_1$ and $y_n=x_n-x_{n-1}$ for $n\geq 2$. Then $(y_n)_{n=1}^\infty$ is a basis for $X$. It is clear ...
4
votes
1answer
29 views

On existence of invariant subspace of continuous linear operator on Banach space such that $\{S(x): S \in (T)'\}=X $ for some $x$

Let $X$ be a Banach space , $T$ be a continuous linear operator on $X$ such that $\exists x \in X$ such that $\{S(x): S \in (T)'\}=X $ , where $(T)'$ is the commutant of $T$ , then I can show that ...
2
votes
1answer
37 views

Classification of subsymmetric basic sequences

I just encountered the following statement: any subsymmetric basic sequence is either weakly null or equivalent to the unit vector basis of $\ell_1$. I see the case in which it is equivalent to the ...
1
vote
0answers
36 views

Existence of a measurable bijection $f:X \to X^2$

Let $(X,\Sigma)$ be a measurable space, where $X$ is an infinite set, and denote by $(X^2,\Sigma^2)$ its product space. Under which conditions it is true that there exists a measurable bijection $f:X ...
2
votes
2answers
45 views

Question on operator theory classes of operators on Hilbert spaces

I was recently tackled by this in my class on operator theory dealing with operators on Hilbert spaces: We are to find and prove the inclusion relations between the classes of operators: ...
3
votes
1answer
43 views

Algebra of compact operators on $\ell_p$

Are the algebras of compact operators $K(\ell_p)$ and $K(\ell_q)$ isomorphic as Banach algebras for $1\leq p<q<\infty$?
1
vote
1answer
45 views

$Y$ be real NLS ; if there is a Banach space $X$ such that there is a continuous linear open mapping from $X$ to $Y$ then is $Y$ a Banach space?

Let $Y$ be a real normed linear space ; if there exist a Banach space $X$ such that there exist a continuous linear open mapping from $X$ to $Y$ then is $Y$ a Banach space ?
0
votes
1answer
26 views

$X$ be Banach space , $T \in \mathcal B(X)$ be an open map , $Y$ be a closed linear subspace of $X$ ; is the restriction of $T$ on $Y$ an open map?

Let $X$ be a Banach space , let $T$ be a continuous open linear map from $X$ to $X$ , let $Y$ be a closed linear subspace of $X$ , then is $T_o$ , the restriction of $T$ on $Y$ , is an open map ? ...
2
votes
0answers
60 views

Looking for a collection of applications of the Open mapping theorem

I am looking for various applications of the open mapping theorem (https://en.wikipedia.org/wiki/Open_mapping_theorem_(functional_analysis) ) other than only the closed graph theorem and that if two ...
0
votes
2answers
53 views

A continuous linear injection from $L^\infty$ into a separable subspace

Can there be a continuous linear injection of $L^\infty$ into one of its closed, separable subspaces? (Note: I am not requiring that injection to be surjective, nor to have closed range). ...