A remarkable example of unbounded and uncountable family of linear functionals on(Mn (R), d)

This paper cites an example of family of linear functional which is as whole bounded at some points of the metric spaces (M_n(R), d) but for any neighborhood of that point there exists a sequence of points such that the family as whole is unbounded at every point of that sequence. For this Baier’s category theorem has been applied. The concept of equicontinuity of families of linear maps has been applied also to reach the result. Now this whole situation can also be thought in terms of Eigen values of matrices and possibly it can throw light on some interesting and useful properties of Eigen values of matrices.  Suppose there is some A M_n(R) such that, the set  (A) = { T_P(B) (A) | P  { R_n(x) } and B G } is bounded set in  ,then for any  0 ,   a sequence {A_n} in B_d (A, ) such that A_n A, for all n  ,the  (A_n) is unbounded subset of  . Where let, {R_n(x)} ≡ The set of polynomials having degree n with real coefficient}, G  The set of real matrices, For each polynomial P { R_n(x)} , we define T_P(B) : M _n(  )     as T_P(B) (A) the trace of A. P(B) ,for each P { R_n(x)} and B  G. T_P(B)  is linear map which is continuous on (M_n (R) , d), where d is the Euclidean metric on M_n (R).