Hi:

First, I must make some definitions, following Neal H.

McCoy, The Theory of Rings.

Def.1: A vector space V over a division ring D is a

D-module with the additional requirement that if 1

is the unity of D and x \in V, then x1 = x.

Def.2: Let k be a positive integer. Then the ring R of

linear transformations is said to be k-fold

transitive if given any ordered set x_1, x_2,...,x_k

of k linearly independent vectors and any ordered

set y_1, y_2,...,y_k of k arbitrary vectors, there

exists a \in R such that x_i a = y_i (i=1,2,...,k).

If R is k-fold transitive for every positive

interger k, R is said to be a dense ring of linear

transformations.

Now, I have this theorem:

Theor.1: Let T be the ring of all linear

transformations of a vector space V with a denumerable

basis. Then...

At a certain point in the proof, he says: Since Ve is

now assumed to have infinite dimension, it has a

denumerable basis X = {x_1, x_2,... } [because V has a

denumerable basis; e \in T; that is, Ve is a subspace

of V].

[T being the ring of ALL linear transformations of V, T

is dense.]

Further on he says: Let Z = {z_1, z_2, ...} be a basis

of V. Since there exists an element of T which maps the

basis elements z_i (i = 1, 2,...) onto any specified

elements of V, there exists an elemnt d of T such that

Code:

z_i d = x_i (i = 1, 2, ...)

.

Here is what I do not understand. Def.2 says that

given any ordered set x_1, x_2,..., x_k of k lin. ind.

vectors in V and a set y_1, y_2,..., y_k of vectors in

V, there exists a\in T such that x_i a = y_i for i = 1,

2, ..., k, and that this is true for any positive k.

An entirely different thing is to say that given a set

x_1, x_2,... of lin. ind. vectors and a set y_1,

y_2,... there is a\in T such that x_i a = y_i for i =1,

2,... That is to say, now he is saying T is w-fold

transitive, where w is omega, the set of natural

numbers. Any suggestion will be welcome.