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Results tagged with banach-spaces
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user 23141
A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.
22
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1
answer
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Who proved that $l^1$ and $L^1[0,1]$ are not isomorphic?
$l^1$ has the Schur property (every weakly convergent sequence is norm convergent) and $L^1[0,1]$ does not, so the two spaces cannot be isomorphic.
Is this folklore, or is it credited to someone? (Al …
- 42.8k
24
votes
2
answers
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Unique predual of a Banach space
Suppose $E$ is a dual Banach space whose predual is unique, and $E_0$ is a codimension 1 weak* closed subspace of $E$. Is the predual of $E_0$ necessarily unique?
Okay, I will reveal the motivation. …
- 42.8k
16
votes
6
answers
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Finding closed subspaces whose sum isn't closed
Let $V_0$ be a closed infinite-dimensional subspace of a Banach space $V$ such that the quotient $V/V_0$ is also infinite-dimensional. Is it always possible to find a closed subspace of $V$ whose sum …
- 42.8k
8
votes
2
answers
360
views
$l^1$ versus $l^2$
Is there an elementary proof of this Banach space fact?
If the Banach space $V$ is linearly isomorphic to $l^1$, then it does not isometrically contain euclidean spaces of arbitrarily large finite …
- 42.8k
23
votes
9
answers
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Nonseparable counterexamples in analysis
When asking for uncountable counterexamples in algebra I noted that in functional analysis there are many examples of things that “go wrong” in the nonseparable setting. But most of the examples I'm t …