Vector

Vector parameters in uncountably infinite-dimensional spaces

Vector parameters in uncountably infinite-dimensional spaces
  1. Can a vector space have infinite dimension?
  2. Can a vector space have uncountable basis?
  3. Which of the following is an infinite dimensional vector space?
  4. Can a vector space have infinite vectors?

Can a vector space have infinite dimension?

Not every vector space is given by the span of a finite number of vectors. Such a vector space is said to be of infinite dimension or infinite dimensional.

Can a vector space have uncountable basis?

In particular some naturally occurring vector spaces can be shown to have an uncountable basis using the axiom of choice. And we can show that this use of the axiom of choice is in fact necessary.

Which of the following is an infinite dimensional vector space?

The Lp spaces are infinite dimensional vector spaces.

Can a vector space have infinite vectors?

A space is infinitely dimensional, if it has no basis consisting of finitely many vectors. By Zorn Lemma (see here), every space has a basis, so an infinite dimensional space has a basis consisting of infinite number of vectors (sometimes even uncountable).

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