I am trying to understand uniform convergence and I almost understand the concept from the series of functions. But the I came up with the question that I do not know is it logical or does it have a correct answer? we can think of series as a special for of sequences I guess. And we are able to talk about uniformly convergence in series my question is if we have matrices (a_nk) (with n row and k column) and let us consider each row as a sequence and let this matrix has infinity row and column. Then we can talk about convergence for each sequence how about uniform convergence how do we define uniform convergence in that case?
thanks.
Note that uniform convergence only makes sense if you have (at least) two variables. For example, a sequence of functions $f_n(x)$ can converge uniformly to $f(x)$, but a sequence of numbers $a_n$ can not. In your case, you can imagine each row of your matrix $(a_{n,k})=(a_1,a_2,a_3,\ldots)^T$ as a function $a_n:\mathbb N\to\mathbb R$, such that $a_{n,k}=a_n(k)$. In this sense the rows of your matrix form a sequence of functions, and then you can define uniform convergence exactly as explained above, that is, $a_n$ converges uniformly to some function $a:\mathbb N\to\mathbb R$, if $\sup_{k\in\mathbb N}|a_n(k)-a(k)|\to0$ as $n\to\infty$.