Uncountable linearly independet family in KNK^\mathbb{N}

Let K be a field. Consider the vector space KN of K-sequences. Is there an uncountable linearly independent set of vectors in this vector space? If Yes, can you name it explicitely? Does this work for modules as well?

Answer

An example I like is this: Say two sets of natural numbers are almost disjoint iff they are infinite but their intersection is finite. Suppose F is an uncountable family of almost disjoint sets. Then the collection of characteristic functions of the sets in F is an example. (If A is a subset of N, then its characteristic function χA is the infinite sequence of 0s and 1s whose n-th entry is 1 iff nA.)

Indeed, if A1,,An,B are distinct sets in F, then there are infinitely many nB but not in iAi, so any linear combination of the χAi is zero at all these entries, but χB is not.

To see that F exists, a cute argument is to note that it suffices to find such a family but made up of subsets of Q. Now, to each real r assign an increasing sequence Ar of rationals converging to r, and take F={ArrR}.


Another, more sophisticated example, uses the concept of independent family. A collection I of subsets of N is called an independent family iff for any distinct A1,,Ak,B1,,Bn in I, we have that the set A1Ak(NB1)(NBn) is infinite.

Again, it is easy to check that if I is an independent family, then the collection of its characteristic functions is linearly independent.

As with the almost disjoint family from above, we can show that there is an independent family of the same size as the reals. But the argument is more involved; a sketch follows: As before, if X is countable, it suffices to find an independent family as required, but consisting of subsets of X (so we replace each NBi with XBi, of course).

Take X={(a,A)aN is finite, and AP(a)}. Check that X is countable.

Now, given SN, let tS={(a,A)XaSA}. The claim is that I={tSSN} is as wanted.

To see this, check first that the assignment StS is injective.

To show that I satisfies the stronger independence requirement, suppose X1,,Xn,Y1,,Ym are infinite, pairwise distinct, subsets of N. Write A=tX1tXn(XtY1)(XtYm). For each i,j with 1in and 1jm let αijXiYj (that is, αij is in either Xi or Yj, but not in both). Show that for any finite F{αij1in,1jm} there is an F such that (F,F)A. This gives the result.


Both these examples, using almost disjoint families, and using independence, are classical applications of infinitary combinatorics to analysis (in this case, to infinite dimensional linear algebra, that is usually studied in the setting of functional analysis). I once asked for the details of the sketch above on a homework problem.

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Source : Link , Question Author : Community , Answer Author : Andrés E. Caicedo

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