Question 36941
YOUR QUESTION IS NOT COMPLETE,BUT FROM WHATEVER IS GIVEN ,I TAKE IT AS FOLLOWS
1.YOU ARE GIVEN A VECTOR SUBSPACE IN T WITH ADDITION AND MULTIPLICATION DEFINED BY SPECIAL OPERATIONS....OK....THEN WE HAVE TO USE ONLY THOSE SPECIAL OPERATIONS TO FIND SUM OR PRODUCT IN THIS SUB SPACE..FOR EX.IF DEFINITIONS ARE AS FOLLOWS THEN

t1=(X1,Y1)		t2=(X2,Y2)	
t1+t2=[{X1^(1/3)+X2^(1/3)}^3,{Y1^(1/3)+Y2^(1/3)}^3]			
C*t1=(C^3*X1,C^3*Y1)	ETC...
WE FOLLOW THOSE DEFINITIONS TO DO OPERATIONS IN T..
IT IS ASSUMED THAT THOSE OPERATIONS ARE CLOSED ,ASSOCIATIVE ETC...IN ACCORDANCE WITH REQUIREMENTS OF VECTOR SUBSPACES OR WE  NEED TO PROVE  IT.

2.IF T MAPS ON TO W OR IS  TRANSFORMED IN TO W ,TO GET THE TRANSFORMATION INTO W FROM T WE USE THE ABOVE OPERATIONS.
THAT IS IF AS GIVEN ABOVE...t1+t2=w1 =(P1,Q1) SAY THEN WE TAKE
   t1+t2=[{X1^(1/3)+X2^(1/3)}^3,{Y1^(1/3)+Y2^(1/3)}^3]	=w1=(P1,Q1)
OR...P1={X1^(1/3)+X2^(1/3)}^3 AND Q1 ={Y1^(1/3)+Y2^(1/3)}^3...ETC...	
3. BUT ONCE WE ARE IN W SPACE ,AS STIPULATED IN THE PROBLEM,
IF W1=(P1,Q1),W2=(P2,Q2)....WE TAKE IT THAT 
W1+W2=(P1+P2,Q1+Q2)...ETC....

HOPE IT IS CLEAR AND THIS IS WHAT YOU WANTED.