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I'm working on the beginning of a book on Banach lattices, and it wants me to show that $C^1(0,1)$ is a Banach Lattice, with the norm $\|f\|=\|f'\|_\infty+|f(0)|$ and the order $f\le g$ if $f(0)\le f(1)$ and $f'\le g'$ pointwise.

I've shown its a Banach space, but I'm having trouble showing it is a lattice. In particular, I am trying to figure out what $f\lor g$ should be. The naive guess of $\max\{f,g\}$ isn't in the space, as it may not be differentiable, as per Is the max of two differentiable functions differentiable?

The "soft max" function listed has the problem of lack of uniqueness, as its really a family of functions that approaches the max. Am I missing something? The question comes from Banach Lattices by Peer Meyer-Nieberg, page 11, exercise 1.1.E3

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Let $h'(x) := \max(f'(x), g'(x))$. $h'$ is continuous. Get a primitive function $h$ with $h(0) = \max(f(0), g(0))$. This should do it.

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  • $\begingroup$ thanks, forgot I could max the derivatives and work there! $\endgroup$
    – Alan
    Commented Jul 26, 2016 at 7:17

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