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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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 ...
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16 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 ...
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34 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
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1answer
23 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 ...
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54 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. ...
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40 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 ...
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1answer
34 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 ...
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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?
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64 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
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1answer
256 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 ...
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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 ...
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34 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
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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 $n \times n$ ...
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1answer
41 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 ...
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66 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 ...
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1answer
62 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 ...
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1answer
53 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 $\mathscr{X}$, ...
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1answer
61 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 ...
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49 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 ...
4
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65 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 ...
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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?
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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 $X$?
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41 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 $P(X)$ ? ...
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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 ...
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1answer
36 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
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1answer
30 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
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1answer
38 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 ...
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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 ...
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2answers
47 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: ...
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44 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 $1\leq p<q<\infty$?
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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 ?
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1answer
28 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 ? ...
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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 ...
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2answers
56 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). ...
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1answer
28 views

Is range of completely continuous of bounded set finite dimensional set?

Define of completely continuous operator : $L$ is continuous operator and map bounded set to relatively compact set , then $L$ is completely continuous operator. Let $\Omega$ is a bounded set of ...
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38 views

If every absolutely convergent series is convergent then $X$ is Banach

Show that A Normed Linear Space $X$ is a Banach Space iff every absolutely convergent series is convergent. My try: Let $X$ is a Banach Space .Let $\sum x_n$ be an absolutely convergent series ...
0
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1answer
38 views

$C^{1}[0,1]$ is not Banach under $\|\cdot\|_{\infty}$ [duplicate]

This is a curiosity from a reading a text that offered no proof. Why is $(C^{1}[0,1], \|\cdot\|_{\infty})$ not Banach?
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1answer
33 views

Goldstine theorem

Given the embedding $j:X\to X''$ defined by, $$j=(x\mapsto(\phi\mapsto\phi(x)))\,,$$ according to my interpretation of the wikipedia page, Goldstine theorem says the following: ...
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20 views

Lower Bound on Norm of Sum of Linearly Independent Vectors in a Banach Space

Suppose X is a Banach space and $x_1,...,x_n$ are linearly independent vectors. Can we find some sort of lower bound for $||\sum x_i||$? What about if we restict ourselves to normalised vectors?
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40 views

The closure of the image of the unit sphere

The following seems very credible to me but is it correct? If $E$ and $F$ are Banach spaces, $T:E\to F$ linear and continuous and $\epsilon>0$, $$\overline{TB_E}\subseteq(1+\epsilon)TB_E\,,$$ ...
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0answers
34 views

Dense surjection into double dual

Since not all Banach spaces $E$ are reflexive, $$\{(E^\ast\ni f \mapsto f(x)): x\in E\}$$ is not necessarily the whole of $E^{\ast\ast}$. However, is it always dense in $E^{\ast\ast}$?
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1answer
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Convergent series which is not absolutely convergent in $(X,\|\cdot\|)$

Let $(X,\|\cdot\|)$ be a normed linear space. Recall from prior results that $(X,\|\cdot\|)$ is Banach $\iff$ any absolutely convergent series in $(X,\|\cdot\|)$ converges. (a) Give an example of ...
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1answer
36 views

Continuous inculsion of the dual of continuous included Banach spaces

If $B$ and $C$ are Banach spaces and $B \subset C$ with the inclusion being continuous. If it true that the set of continuous linear functionals on $C$, $C'$, is continuous included in the set of ...
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Contractive Operators on Compact Spaces

Suppose that $T: M \to M$ is a compact contractive Operator on a nonempty compact subset $M$ of a complete metric space $X$. Show that $T$ has a unique fixed point. Further show that the sequence ...
3
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1answer
30 views

Uniform lower bound of sequence of linear maps on a Banach space

Suppose a sequence of $T_j:X\to\mathbb{R}$, with $X$ Banach, has the following property: $$\forall j:\|T_j\|\geq c>0$$ Then we have for all $n$, $$\exists x\in X\setminus \{0\}:\forall j\leq ...
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1answer
173 views

Weak$^*$-convergence of vector-valued measures implies weak$^*$-convergence in $X^*$?

Let $K$ be a compact Hausdorff space and $X$ be a Banach space. By the Riesz-Singer representation theorem, we know that there exists a linear isometry from $C(K,X)^*$ onto $rcabv(K,X^*)$, the Banach ...
2
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1answer
29 views

Norm operators bounded below implies almost uniform lower bound

I have a hard time proving (or disproving) the following statement about continuous linear operators: $$(\exists c>0:\forall j:\|T_j\|\geq c)\Rightarrow(\exists\delta>0:\forall n:\exists x\in ...
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1answer
45 views

A question in the proof of $C(X)$ is not a dual space of a Banach space.

Let $X$ be a non-singleton compact connected space.I want to show that $C_{\mathbb{R}}(X)$ is not the dual space of a Banach space,$\mathbb{R}$ is real number field. I already know that, extreme ...
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65 views

Uniform limit of m-homogeneous polynomials over compact subsets of a Banach Space

I am trying to solve problem 1.2.A from Mujica's book "Complex Analysis in Banach Spaces". We denote by $\mathcal{P}_a(^mE;F)$ the space of all $m-$homogeneous polynomials from $E$ into $F$, i.e, the ...
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1answer
37 views

Degree of infinite dimensinal antipodal map

Let $E$ is a infinite dimensional Banach space and $\Omega=\{x\in E: ||x||=1\}$ . $L:\Omega\rightarrow\Omega,L(x)=-x$, how to compute the degree of $L$? In fact ,I just know that the algebra define ...