Banach spaces of infinite dimension do not have a countable Hamel basis
A Banach space^{} of infinite^{} dimension^{} does not have a countable^{} Hamel basis^{}.
Proof
Let $E$ be such space, and suppose it does have a countable Hamel basis, say $B={({v}_{k})}_{k\in \mathbb{N}}$.
Then, by definition of Hamel basis and linear combination^{}, we have that $x\in E$ if and only if $x={\lambda}_{1}\cdot {v}_{1}+\mathrm{\dots}+{\lambda}_{n}\cdot {v}_{n}$ for some $n\in \mathbb{N}$. Consequently,
$$E=\bigcup _{i=1}^{\mathrm{\infty}}(\mathrm{span}{({v}_{j})}_{j=1}^{i}).$$ |
This would mean that $E$ is a countable union of proper subspaces^{} of finite dimension (they are proper because $E$ has infinite dimension), but every finite dimensional proper subspace of a normed space^{} is nowhere dense, and then $E$ would be first category. This is absurd, by the Baire Category Theorem.
Note
In fact, the Hamel dimension of an infinite-dimensional Banach space is always at least the cardinality of the continuum^{} (even if the Continuum Hypothesis fails). A one-page proof of this has been given by H. Elton Lacey[1].
Examples
Consider the set of all real-valued infinite sequences $({x}_{n})$ such that ${x}_{n}=0$ for all but finitely many $n$.
This is a vector space^{}, with the known operations^{}. Morover, it has infinite dimension: a possible basis is ${({e}_{k})}_{k\in \mathbb{N}}$, where
$${e}_{i}(n)=\{\begin{array}{cc}1,\hfill & \text{if}n=i\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}$$ |
So, it has infinite dimension and a countable Hamel basis. Using our result, it follows directly that there is no way to define a norm in this vector space such that it is a complete metric space under the induced metric.
References
- 1 H. Elton Lacey, The Hamel Dimension of any Infinite Dimensional Separable^{} Banach Space is c, Amer. Math. Mon. 80 (1973), 298.
Title | Banach spaces of infinite dimension do not have a countable Hamel basis |
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Canonical name | BanachSpacesOfInfiniteDimensionDoNotHaveACountableHamelBasis |
Date of creation | 2013-03-22 14:59:12 |
Last modified on | 2013-03-22 14:59:12 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 21 |
Author | yark (2760) |
Entry type | Result |
Classification | msc 46B15 |