Banach spaces of infinite dimension do not have a countable Hamel basis
Let be such space, and suppose it does have a countable Hamel basis, say .
This would mean that is a countable union of proper subspaces of finite dimension (they are proper because has infinite dimension), but every finite dimensional proper subspace of a normed space is nowhere dense, and then would be first category. This is absurd, by the Baire Category Theorem.
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.
Consider the set of all real-valued infinite sequences such that for all but finitely many .
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.
- 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|
|Date of creation||2013-03-22 14:59:12|
|Last modified on||2013-03-22 14:59:12|
|Last modified by||yark (2760)|