# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 $\liminf_n \|x_n\| \geq \|x\|$ ...
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 ...