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I am struggling to proof that $$||A||_1=\max_{||x||_1=1}||Ax||_1$$ I am suppose to start by showing if $$||x||_1=1||$$ then $$||Ax||_1 \leq \max_j\sum_{i=1}^m|A_{ij}|$$

I thought of maybe starting with $$||Ax||_1 \leq ||A||_1||x||_1$$ $$||Ax||_1\leq||A||_1\sum_{j=1}^n|x_j|$$ But after that I have no clue where to go?

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    It would help if you post your definition of $\|A\|_1$. – 2017-02-01
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    I am proving the definition of a 1-norm. Which is $||A||_1=\max_{||x||_1=1}||Ax||_1$ – 2017-02-01
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    Maybe i should state that A is an mxn matrix and x is a vector with n elements – 2017-02-01
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    See [here](http://fourier.eng.hmc.edu/e161/lectures/algebra/node12.html), just after the properties of a matrix norm. – 2017-02-01
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    @daultongray8: One does not *prove* definitions. They are what they are. Surely you have a different definition you're starting with? Maybe $\|A\|=\sup_{x}\frac{\|Ax\|_1}{\|x\|_1}$? – 2017-02-01

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