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Let $F$ be a field, and consider the infinite direct product$$F \times F \times F \times F \times \dots,$$i.e. $\prod_{i=0}^\infty F$, i.e. the direct product of a countable number of copies of $F$. The theory of bases for vector spaces shows that this direct product has a basis, and so is isomorphic to a direct sum of copies of $F$. How many copies of $F$ do I need $($i.e. what is the dimension of this direct product as an $F$-vector space$)$? Is it possible to exhibit this isomorphism between an infinite direct product and an infintie direct sum explicitly?

  • Interesting: this question has appeared few times on M.SE in the last days! Same homework? (See http://math.stackexchange.com/questions/1213082/how-to-calculate-the-dimension-of-an-infinite-direct-product-of-copies-of-a-fiel#comment2476525_1213082 and http://math.stackexchange.com/questions/1217128/the-dimension-of-vector-space-fx) – user26857 Apr 09 '15 at 15:34
  • Admittedly, this is an extra credit homework question, but I looked at the names of the other users who asked the same question and I do not believe they are in my class. Just a coincidence, I suppose. –  Apr 09 '15 at 20:56

2 Answers2

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The dimension is the cardinality $|F|^{\aleph_0}$ according to the Theorem of Erdös-Kaplansky. See MO/49551 for a general formula of the dimension of an infinite family of vector spaces. All this belongs to "set-theoretic linear algebra", and no bases can be explicitly written down, since the axiom of choice plays a central role. Probably there are models of ZF where $\mathbb{Q} \times \mathbb{Q} \times \dotsc$ is no direct sum of copies of $\mathbb{Q}$; hopefully a set-theorist can add a reference for this.

Community
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Martin Brandenburg
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    It wasn't stated in the question or in this answer, so I'll just state for the record that $\dim(\bigoplus_{\Bbb N}F) = \aleph_0$ (in contrast to the direct product), pretty much by the definition of a direct sum. – Ben Grossmann Apr 09 '15 at 00:10
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To see the difference between infinite direct product and infinite direct sum (both involving countable number of copies) see their concrete realisations: The vector space of all polynomials in $X$ over $F$, and the vector space of all formal power series (convergent or not) in $X$ over $F$. Now you can see what forms the basis in each case.

P Vanchinathan
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  • The difference between direct product and direct sum can be directly seen from their construction. 2) Polynomials and power series are no concrete realizations; I would rather say that the underlying vector space of the $F$-algebra $F[x]$ resp. $F[[x]]$ is $F^{\oplus \mathbb{N}}$ resp. $F^{\mathbb{N}}$. Thus, polynomials and power series carry more structure. 3) The notion of convergence doesn't make sense here. 4) You claim that one can see a basis of $F^{\mathbb{N}}$. This is wrong. For example, ${1,e_1,e_2,\dotsc}$ is no basis, it generates sequences which are eventually constant.
    • – Martin Brandenburg Apr 09 '15 at 07:44
  • @Martin Brandenberg: Thanks for the precision, and pointing out the mistake. Perhaps what I said makes sense only as groups: and not as $F$-algebra. That is they are concrete examples of infinite direct produc as groups and infinite direct sum as groups. Is this correct? – P Vanchinathan Apr 09 '15 at 09:55