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Let $A=[a_{ij}]$ be a matrix with $\infty$ rows and $\infty$ columns with entries in a vector space, also $A_n=[b_{ij}]$ a matrix with $\infty$ rows and $\infty$ columns when $b_{ij}=a_{ij}$ for $i,j\leq n$ and $b_{ij}=0$ for $i,j>n$.

Is $A=\sup A_n$ Correct or not? why?

Can introduced me a reference about these matrices?

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    What do you mean by $\sup$ for a matrix??? – 2017-01-08
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    @copper.hat In fact, I need to show $A$ is built by $A_n$. or $A=\lim A_n$ – 2017-01-08
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    In this context, I guess you are referring to Hilbert-Schmidt operator, (See an introduction in https://en.wikipedia.org/wiki/Hilbert%E2%80%93Schmidt_operator) which analogue to 'matrix in infinite dimension'. – 2017-01-08
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    @Lion I put $\|.\|_\infty$ on matrix. – 2017-01-08

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