Actually, in case the norm is consistent, the proof shows more than the thesis; in fact, using the previous lemma, we can replace in the limit definition the left lower bound with the spectral radius itself and write more precisely:
22.
First of all, using Gerschgorin's theorem, I put a bound on the spectral radius of this matrix as 20M where M is the maximum of | a ( x ) | on the entire grid ( I could use the max on [ 0, 1 ] too which would be uniform for all grids ).
23.
To put it another way, f ( A ) \, converges absolutely for every square matrix whose spectral radius is less than the radius of convergence of f around 0 and is uniformly convergent on any compact subsets of \ mathbb { M } _ n ( \ mathbb { C } ) satisfying this property in the matrix Lie group topology.
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