Consider the following problem 5 below.
I am trying to construct the stated function f. I tried many functions one which sends the matrix of the form below to xy. I also tried to send the matrices of the form below to x + y. None of them work.
Recall a left uniform continuous function f on G is defined as follows. First define the left and right translated of f through y by:
$$L_yf(x) = f(y^{-1}x)$$ $$R_yf(x) = f(xy)$$
f is called left (resp. right) uniformly continuous if for every $\epsilon$ there is a neighborhood V of e such that $\|L_yf - f\|_u < \epsilon$ resp $\|R_yf - f\|_u < \epsilon$ for $y \in V$. We are working with uniform continuity here. Does anyone the correct function here ?
