Dual of a direct sum

In most of what follows we shall view the notion of direct sum as defined for subspaces of a vector space ; this avoids the fuss with the identification convention of Section: Direct sums , and it turns out, incidentally, to be the more useful concept for our later work. We conclude, for the present, our study of direct sums, by observing the simple relation connecting dual spaces, annihilators, and direct sums. To emphasize our present view of direct summation, we return to the letters of our earlier notation.

Theorem 1. If and are subspaces of a vector space , and if , then is isomorphic to and to , and .

Proof. To simplify the notation we shall use, throughout this proof, , , and for elements of , , and , respectively, and we reserve, similarly, the letters for and for . (This notation is not meant to suggest that there is any particular relation between, say, the vectors in and the vectors in .)

If belongs to both and , i.e., if for all and , then for all ; this implies that and are disjoint. If, moreover, is any vector in , and if , we write and . It is easy to see that the functions and thus defined are linear functionals on (i.e., elements of ) belonging to and respectively; since , it follows that is indeed the direct sum of and .

To establish the asserted isomorphisms, we make correspond to every a in defined by . We leave to the reader the routine verification that the correspondence is linear and one-to-one, and therefore an isomorphism between and ; the corresponding result for and follows from symmetry by interchanging and . (Observe that for finite-dimensional vector spaces the mere existence of an isomorphism between, say, and is trivial from a dimension argument; indeed, the dimensions of both and are equal to the dimension of .) ◻

We remark, concerning our entire presentation of the theory of direct sums, that there is nothing magic about the number two; we could have defined the direct sum of any finite number of vector spaces, and we could have proved the obvious analogues of all the theorems of the last three sections, with only the notation becoming more complicated. We serve warning that we shall use this remark later and treat the theorems it implies as if we had proved them.

EXERCISES

Exercise 1. Suppose that , , , and are vectors in ; let and be the subspaces of spanned by and respectively. In which of the following cases is it true that ?

, , , .
, , , .
, , , .

Exercise 2. If is the subspace consisting of all those vectors in for which , and if is the subspace of all those vectors for which , , then .

Exercise 3. Construct three subspaces , , of a vector space so that but . (Note that this means that there is no cancellation law for direct sums.) What is the geometric picture corresponding to this situation?

Exercise 4.

If , , and are vector spaces, what is the relation between and (i.e., in what sense is the formation of direct sums an associative operation)?
In what sense is the formation of direct sums commutative?

Exercise 5.

Three subspaces , , and of a vector space are called independent if each one is disjoint from the sum of the other two. Prove that a necessary and sufficient condition for (and also for ) is that , , and be independent and that . (The subspace is the set of all vectors of the form , with in , in , and in .)
Give an example of three subspaces of a vector space , such that the sum of all three is , such that every two of the three are disjoint, but such that the three are not independent.
Suppose that , , and are elements of a vector space and that , , and are the subspaces spanned by , , and , respectively. Prove that the vectors , , and are linearly independent if and only if the subspaces , , and are independent.
Prove that three finite-dimensional subspaces are independent if and only if the sum of their dimensions is equal to the dimension of their sum.
Generalize the results (a)-(d) from three subspaces to any finite number.

EXERCISES

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