# 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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### A complex unital algebra which is a Banach space is also a Banach algebra

In my functional analysis class I got stuck on this question on Banach algebras: Let $\mathbb{A}$ be a complex unital algebra (it has a unit i.e. a multiplicative identity) and $\mathbb{A}$ is ...
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### $X$ be Banach space with a proper dense subspace $Y$. Can the identity operator on $Y$ be extended to a continuous function from $X$ into $Y$?

Let $X$ be any Banach space with a proper dense subspace $Y$. Can the identity operator on $Y$ be extended to a continuous function from $X$ into $Y$ ?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, finite....
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### 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.
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### $\{x_n\}$ 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 $\sum_{n=1}^{\infty} f(x_n)$ ...
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### 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 ...
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