On whether or not [formula omitted]( E, F) = [formula omitted]r( E, F) for some classical Banach lattices E and F

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For Banach lattices E and F, L ( E,F) is the space of all continuous linear operators E→ F, L r( E,F) is the vector space of all regular continuous linear operators E→ F which is endowed with the r-norm. This paper concerns the problems: (1) is every continuous linear operator E→F regular? (2) if the answer to (1) is “yes”, there is a further problem: is its operator norm in L ( E,F) equal to its r-norm in L r( E,F)? A series of conclusions is obtained for cases in which each of E and F is one of Banach lattices l p (1≤ p < ∞), l ∞, c 0, c, C[0,1] and C( X).