60 BIRKHOFF AND LANGEE. 



Denoting by the symbol « the fact that each element of the matrix 

 of constants on the right represents a value as large as the largest 

 numerical maximum attained by any element of the matrix on the left, 

 we have, since A(x) and Ufa) are continuous in the closed interval 

 a ^ x fS b, 



Ufa)«{k), 

 A (x) « (a) . 



From this it follows that 



Ufa) = A(x) Ux (x) = (xaa ugj «{nak) 



and integrating 



£7 2 (.r)«(^c*|.r-.r |). 



Likewise U' 3 = AU 2 = I Sfl«tty ) «( 2a- knot \ x — Xq \ ), 



U s (x) « hW |,r ~ ,ro12 ), 

 and similarly for I = 3, 4, . . . , 



/ i 



x — x 



\l\ 



U l+1 (x) « IM - - '- «k 



[nab — a} 



I! I \ l! 



However the infinite series of matrices 



(% Unb^a\ l \ 



a=o 



11 



converges to the matrix 



Hence the series 



k (e ani - a —\). 



Y + 2 Ufa) 



converges uniformly throughout the interval a ^ x ^ b, and since its 

 terms are matrices of continuous functions the matrix Y(x) which it 



