# Tagged Questions

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### The closed graph theorem for Banach spaces isn't true. True?

I'm reading through some functional analysis lecture notes and there the closed graph theorem was stated in the following form: Let $X$ be a Baire locally convex space and $Y$ a Frechet space. If ...
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### Counterexamples for Hölder's inequality when $p$ and $q$ are not conjugate.

HÃ¶lder's inequality shows that, when $$\frac{1}{p} + \frac{1}{q} = 1,$$ and $f\in L^p$ and $g\in L^q$, then $$\Vert f\,g\Vert_1 \le \Vert f \Vert_p \Vert g \Vert_q.$$ Is there an example of this ...
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### Condition to separability of a Banach space.

I am trying to prove the following statement: Let X be a Banach space and $X^{*}$ its topological dual space. If there exists a countable family of functions $(f_{n})_{n} \subset X^{*}$ such that ...
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### Completeness is not preserved under homeomorphism

I am trying to give a counterexample which will make it clear that homeomorphism does not preserve completeness. I have an easy one in mind ($(0,1)$ and $\mathbb{R}$) but I have just thought that ...
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### Counterexample for the Chain rule for the Gateaux-derivative

I'm reading the book of Drabek, Milota - Methods of Nonlinear Analysis, and at page 121, they state: but I can't manage to find such counterexample. For clarity the Gateaux derivative is defined ...
### How to construct an “explicit” element of $(\ell^\infty(\mathbb N))^* \setminus \ell^1(\mathbb N)$? [duplicate]
Possible Duplicate: Nonnegative linear functionals over $l^\infty$ An explicit functional in $(l^\infty)^*$ not induced by an element of $l^1$? Everything is in the title: How to ...