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Let $X_{n}$ be the set of all $n \times n$ matrices of the real numbers. For each $(a_{ij})\in X_{n}$ and $\epsilon > 0$ let $U_{\epsilon}((a_{ij})) = \left\{ (b_{ij})\in X_{n} \,|\, \forall i,j, \,|b_{ij}-a_{ij}|<\epsilon \right\}$.

I need to prove that these sets form a basis for a topology and describe what "familiar space" these look like.

First of all, I am a bit confused about the notation: I am assuming that $(a_{ij})$ and $(b_{ij})$ are referring to matrices in $X_{n}$, and I think, though I'm not sure that $|b_{ij}-a_{ij}|<\epsilon$ means that the difference in the determinants for these matrices should be arbitrarily small. If I'm wrong and you're familiar with this basis/problem, let me know. I tried to contact my prof. to ask him what he meant, but I have still received no reply.

Now, as for how to approach this problem, ideally, I would want to show that any matrix $C = (c_{ij}) \in X_{n}$ is contained in one of the basis elements (i.e., contained in one of the $(b_{ij})$, and then show that any matrix $C$ can likewise be contained in another matrix, which is contained inside the intersection of two of the elements of $U_{\epsilon}$.

My question is, how do I show that one matrix is contained in the other? Also, how do I get a matrix contained in the intersection of two other matrices?

I am extremely confused about even where to begin with this problem!

I tried lookng at the case when $n=2$, then, found that $|(b_{ij})-(a_{ij})| = \begin{vmatrix} b_{11}-a_{11} & b_{12}-a_{12} \\ b_{21}-a_{21} & b_{22} - a_{22} \end{vmatrix} = b_{11}b_{22}-b_{11}a_{22}-a_{11}b_{22}+a_{11}a_{22}-b_{12}b_{21}+b_{12}a_{21}+a_{12}b_{21} - a_{12}a_{21} < \epsilon$.

This did not help me.

I also tried doing that for matrices of arbitrary $n\times n$ size, by doing cofactor expansion across the first row. Still did not help me.

Needless to say I have no idea what I'm doing...could somebody please help me?

Thank you.

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The notation means that the matrices $A$ and $B$ are entrywise close. In other words, your topology is given by the metric $$ d(A,B)=\|A-B\|_\infty, $$ where $$ \|A\|_\infty=\max\{|a_{ij}|:\ i,j=1,\ldots,n\} $$ (where, of course, $a_{ij}$ are the entries of $A$). With the above definitions, $U_\epsilon(A)$ is precisely the ball of radius $\epsilon$ around $A$.

As a comment about notation, it is fairly common to write $(a_{ij})$ to denote a matrix, to emphasize the form of its entries.

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    so are you saying that the topology resembles the topology induced by that metric, but in $n \times n$ dimensions? – 2017-02-21
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    No, I didn't say "resembles". I said that such is precisely the topology you described in your question. – 2017-02-21
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    how do I demonstrate that it fits the definition of a basis, though? – 2017-02-21
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    The sets $U_\epsilon(A)$ are precisely all the open balls in the metric I mention in my answer. – 2017-02-21
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    That is not the usual definition of the matrix infinity norm, but it is the one you want in this case. – 2017-02-21
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    @brennan: I'm not aware of a canonical "matrix infinity norm". It would be a good name for the operator norm, since it would be the analog at$\infty$ for the Schatten $p$-norms. But I have seen so many names attached to so many different matrix norms, that I don't really remember a single one. – 2017-02-21
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    @Martin: yeah, there are a ton of these things. But I think that it's pretty standard to define $||A||_{\infty}$ as the max absolute row sum. https://en.wikipedia.org/wiki/Matrix_norm#Schatten_norms This is all besides the point though, I just threw this up in case some one got confused. – 2017-02-22
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    @brennan: as far as I can tell, the Wikipedia article you linked to uses $\|A\|_\infty $ for the two norms I mentioned: the max and the operator norms. – 2017-02-22