So I'm having trouble understanding the max notation and exactly what it means.

I have a problem I'm working and this is what it is saying

What is the interpretation of this in english, I understand this much.

L1 norm of A = to the max of ||Ax|| when ||x|| = 1 , but what is a good way of thinking about it.

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I've also seen the notation used with arg max, I assume they are the same. I'm just having a hard time understanding the interpretation of this.