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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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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13 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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46 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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38 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 ...
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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 ...
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1answer
40 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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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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1answer
52 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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46 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
33 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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62 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 ...
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2answers
58 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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2answers
43 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 ...
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1answer
39 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
35 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: $$\overline{jB_X}^{...
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24 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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1answer
43 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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35 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
40 views

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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82 views

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 ...
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1answer
31 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 n:|...
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1answer
223 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
31 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
51 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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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 ...
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1answer
65 views

On the Banach–Alaoglu theorem: is the unit ball of an equivalent norm also weak-* compact?

Suppose that $E$ is a Banach space and let $E^*$ denote its dual space with canonical norm $\lVert\bullet\rVert_{E^*}$. Suppose that $\lvert\bullet\rvert_{E^*}$ is an equivalent norm on $E^*$. The ...
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showing that $f_n(x)= x^n$ is no Cauchy Sequence

Considering the space $C^0([0,1])$ of continuous functions on $[0,1]$, with the norm $||f|| = \max_{x \in [0,1]} |f(x)|$ I have to determine whether $f_n(x) = x^n, n \in \mathbb N$ is a cauchy ...
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1answer
58 views

if Banach space $X\cap Y$ is dense both in $X$ and $Y$, can we have $(X\cap Y)^*=X^*+Y^*$?

Let $X$ and $Y$ be sub-spaces of a large vector space, and both formed Banach spaces with associated norm. If $X\cap Y$ with norm $\|u\|_{X\cap Y}=\|u\|_X+\|u\|_Y$, is dense in $X$ and $Y$ , it's ...
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45 views

Completely continuous map is not homotopy with antipodal map

Define of completely continuous operator : $L$ is continuous operator and map bounded set to relatively compact set , then $L$ is completely continuous operator. Now, $E$ is a infinity dimensional ...
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1answer
37 views

Separation Hahn Banach theorem in vector Banach lattice

Let $X$ be a vector Banach lattice. Let $C$ be a closed cone of positive elements in $X^+$ and let $0\leq x\in X-C$. Q: Does there exists any bounded positive linear functional $f$ on $X$ by which $...
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Tensor product of $L^1([0,1],\omega)$ with some Banach space

We know that for any Banach space $E$ if $\hat{\otimes}$ denotes projective tensor product then $L^1([0,1])\hat{\otimes} E$ is isomorphism isometric with $$L^1([0,1],E)=\{f:[0,1]\to E:\ \ \int_0^1\|f(...
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1answer
82 views

$x^*\circ f:G\rightarrow \Bbb{C}$ is analytic. Show $f$ is analytic.

This is an exercise in Conway's 《A course in Functional Analysis》. X is a complex Banach space. $G$ is an open set in the complex plane and $f:G\rightarrow X$ is a function such that for each $x^{*}$ ...
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20 views

Relation between homomorphisms and Lipschitz functions

Given a Riemannian (say, connected) manifold $(M,g)$ one can produce the metric via a formula $d(x,y)=\inf_{\gamma}l(\gamma)$ where $\gamma$ is piecewise smooth curve joining $x$ and $y$. There is ...
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2answers
73 views

Given $S \in B(Y^{*}, X^{*})$, does there exist $T\in B(X,Y)$ such that $S=T^{*}$?

Let $X, Y$ be Banach spaces, $S \in B(Y^{*}, X^{*})$. Does such operator $T \in B(X, Y)$ exist so that $T^{*}=S$? I suppose that the answer should be - no. Are there any hints that might help in ...
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1answer
41 views

Complex of banach spaces is exact if and only if its dual is exact

Let's consider two complexes of Banach spaces: $ X \rightarrow Y \rightarrow Z$, with the maps $S: X \rightarrow Y$, $T: Y \rightarrow Z$. The dual complex looks like $Z^{*} \rightarrow Y^{* } \...
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1st isomorphism theorem on linear transformations between Banach spaces

First isomorphism theorem: Let V,W be vector spaces and $f:V\to W$ be modules homomorphism then $V/ker(f)\cong Im(f)$. On the other hand, by an exercise in Folland's "Real Analysis" book (chapter 5, ...
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1answer
68 views

If $u\in L^2(U)$ and $u>0$, how to show $u\ln u\in L^2(U)$?

$U$ is a bounded open subset of $R^n$. If $u\in L^2(U)$ and $u>0$, how to show $u\ln u\in L^2(U)$ ?
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1answer
38 views

Showing codimension of subspace of C[0,1] equals 1

Show that $\overline{\{f∈C^1[0,1]:f(0)=0\}}$ as a subspace of $C[0,1]$ has codimension 1. Attempt: define $T:C[0,1]\to$ $\Bbb{R}$ by $T(f)=f(0)$. $T$ is a surjective continuous linear transfomation ...
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17 views

Differentiation of a continuous bilinear form

Let $E_1, E_2$ and $F$ be $3$ Banach spaces. Let $B: E = E_1 \times E_2 \to F$ be a continuous bilinear form. Show that $B$ is differentiable at every point $a = (a_1,a_2) \in E$ and its differential ...
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Does it necessarily follow that $C^{n, \gamma}(\overline{X})$ is a Banach space? [duplicate]

Let $X$ denote an open subset of $\mathbb{R}^n$. Suppose $n \in \{0, 1, \dots\}$, $0 < \gamma \le 1$. Does it necessarily follow that $C^{n, \gamma}(\overline{X})$ is a Banach space?
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Weak Convergence Inequality

Let $X$ be a Banach space and $X^*$ its dual space. a) If $\left\lbrace x_n\right\rbrace$ converges weakly to $x$ in $X$, then $\sup_n \|x_n\| < \infty$ and $\liminf_n \|x_n\| \geq \|x\|$ ...
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2answers
48 views

Showing a set is closed and nowhere dense in a Banach space

Let $X$ and $Y$ be Banach spaces, $T_j \in L(X,Y)$ for each $j$ and let $E_n = \left\lbrace x \in X: \sup_{j \geq 1} \|T_jx\| \leq n\right\rbrace$. Show $E_n$ is closed for each $n$ If ...
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1answer
80 views

Closed Graph Theorem Application

I'm having trouble working out the proof for this problem. Let $T:L^2(X)\to L^2(X)$ be a linear map such that there is another linear map $T^*:L^2(X)\to L^2(X)$ with $\langle Tu,v\rangle=\langle u,T^*...
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1answer
40 views

How to show that a Banach space is a subspace of another Banach space?

I'm having some trouble showing that the following Banach space is a subspace of $\ell^1(\mathbb{N} )$ $\ell^1_w(\mathbb{N} ) = \{ \{x_k\}^\infty_{k=1}| x_k \in \mathbb{C}, \sum_{k=1}^{\infty} |x_k|\...