Hamel in Banach Space

Question

A Hamel basis is a subset B of a vector space V such that every element v ∈ V can uniquely be written as

${\displaystyle v=\sum _{b\in B}\alpha _{b}b}$ with $α_b ∈ F$, with the extra condition that the set

${\displaystyle \{b\in B\mid \alpha _{b}\neq 0\}}$ is finite. This property makes the Hamel basis unwieldy for infinite-dimensional Banach spaces.

Is this sufficient for the question. Thanks

https://math.stackexchange.com/questions/599903/no-infinite-dimensional-f-space-has-a-countable-hamel-basis?rq=1 — Sumanta, Sep 06 '20 at 11:12
https://math.stackexchange.com/questions/3171921/is-there-a-notion-of-a-continuous-basis-of-a-banach-space?rq=1 — Sumanta, Sep 06 '20 at 11:12

score 2 · Answer 1 · answered Sep 06 '20 at 12:34

Almost always, when using a Banach space, we are interested in the topological properties related to the norm. A Hamel basis is not useful in such situations.

In Banach space, there is a more useful notion of a Schauder basis, where we allow infinite linear combinations $$ x = \sum_{k=1}^\infty \alpha_k x_k $$ but we require of course convergence in the norm.

Hamel in Banach Space

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