Bishop-Phelps theorem

We are now ready to apply the Bishop-Phelps theorem to the Lieb functional Fh. We take e = 1, n0 G S and let —vk correspond to the F-bounded functional L0 of the theorem. According to the theorem we can then find a tangent functional — wk (the Le of the theorem) such that... 

Theorem 12 (Bishop-Phelps I). Let F B —<> 1Z be a lower semicontinuous convex functional on a real Banach space B. The functional F can take the value +oo but not everywhere. Then the continuous tangent functionals to F are B -norm dense in the set of F-bounded functionals in B. ... 

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