Tagged Questions

1answer
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Sums of special vectors

Let $v$ be a vector obtained by taking a sum of $k$ vectors the of the form $(0,0,\ldots,0, -n, *,*,\ldots,*)$, where $"*"$ stands for either $0$ or $1$, and the position of the $-n$ entry can vary ...
3answers
49 views

Not a basis for $l^\infty$ then what is it?

We know that $l^\infty$ has not a Schauder basis and its Hamel basis is uncountably infinite. Let $e_n=(e_{n1}, e_{n2},...)$ (for each $n\in \mathbb{N}$) s.t. $e_{nj}=0$ when $n\neq j$ and ...
1answer
52 views

1answer
491 views

For Banach space there is a compact topological space so that the Banach space is isometrically isomorphic with a closed subspace of $C(X)$.

I want to prove that for Banach space V there is a compact topological space $X$ so that $V$ is isometrically isomorphic to a closed subspace of $C(X)$-continuous function on a (compact) topological ...
2answers
247 views

When is $\mathcal{L}_\infty$ a vector space?

Suppose that $X$ is a subset of a Hausdorff topological space, and $Z\subseteq X$ a member of the family of Borel subsets on $X$. Let $B(Z)$ the set of bounded functions on $Z$, equipped with the ...
2answers
3k views

Connections between metrics, norms and scalar products (for understanding e.g. Banach and Hilbert spaces)

I am trying to understand the differences between  \begin{array}{|l|l|l|} \textbf{vector space} & \textbf{general} & \textbf{+ completeness}\\\hline \text{metric}& \text{metric ...
1answer
979 views

Vector, Hilbert, Banach, Sobolev spaces

Trying to wrap my head around all these different spaces. Which one is the most general? Can you summarize the differences between them? Is there a notable space that I missed?