Thread: About the continuity of coordinate projection

I have attempted to solve this problem; however I couldn't do it.

Please, could someone give me a solution or a partial solution?

Thank you.

Consider the vector space of all sequence of complex numbers. Let ||.|| be a norm in such space. Why at most a finite number of coordinate projections can be continuous (or b.ounded) in the norm ||.||?

Without loss of generality assume that every projection is bounded with a constant K_i>0, which means that for all x=(x_1,x_2,...)
|| x_i e_i || <= K_i || x ||

where e_i is the i-th unit vector.

Then in particular for the vector x, defined by x_i:=i*K_i/|| e_i ||

|| x_i e_i ||=i*K_i <= K_i || x ||
=> || x || >= i
for all i
which is impossible.

Thank you very much for your help. This solved my problem.