Suppose V is finite-dimensional with dimV>0 and suppose W is infinite-dimensional. Prove that L(V,W) is infinite-dimensional.
Uhh, obviously?
Let v∈V and let w1,… be a sequence of vectors in W such that w1,…,wm is independent for all m (see 2a/14 for the existance of such a sequence).
Define Tj(v)=wj and define the rest of Tj so that Tj∈L(V,W). To show T1,… is an independent sequence in L(V,W) consider
a1T1+⋯+amTm=0
But
a1T1v+⋯+amTmv=a1w1+⋯+amwm=0
Implies each aj=0 since w1,…,wm are independent. Thus T1,… is an independent sequence implying L(V,W) is infinite-dimensional by 2a/14.