Let M_n(R) be the n x n matrices over the reals R. Define a norm || || on M_n(R) by ||A||= sum of absolute values of all the entries of A. Further define a new norm || ||* by ||A||* = sup{||AX||/||X||, ||X||!=0}.(adsbygoogle = window.adsbygoogle || []).push({});

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1. M_n(R) under || ||* is complete.

2. If ||A||<1, then I-A is nonsingular, where I is the identity matrix.

3. The set of nonsingular matrices in M_n(R) is open.

4. Find ||B||*, where B is 2x2 and b_11=1, b_12=2, b_21=3, b_22=4.

There is a series of over 10 questions on the norm || ||. I've solved most of them but I've been stuck on (have no clue for) these ones above for a week.

I'd appreciate any hints.

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# Matrix Banach Space

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