Renormings of L p (L q )

V-Proximal Trustworthy Banach Spaces

“…For a positive measure space (Ω, Σ, μ), we denote by L p , p ∈ ½1,∞Þ, the Banach space L p ðΩ, Σ, μÞ with its canonical norm kxk = ð Ð Ω jxðwÞj p dμðwÞÞ 1/p . We recall the following result from Theorem 1.1 in Section 5.1 in [7] (see also [8]).…”

Iterative Schemes for Nonconvex Quasi-Variational Problems with V-Prox-Regular Data in Banach Spaces

“…We notice that the class of spaces satisfying the assumptions of the previous theorem is very large; it contains obviously any Hilbert space and spaces and Sobolev spaces , with ≥ 2 (see Theorem 1.1 in Section 5 in [10,12]) and for more examples and discussions we refer to [10,12]. We close this section with the following two concepts of uniform -prox-regularity for functions and sets (see [13]).…”

Calculus Rules forV-Proximal Subdifferentials in Smooth Banach Spaces

“…The assumption on the norm used in the previous proposition is true for any Hilbert space and for many other Banach spaces (see, e.g., [9,12]). Using Theorem 1.1 in Section 5.1 in [9] we get that all spaces for ≥ 2 satisfy the 2 property with their canonical norm and the same result is also true for the Sobolev spaces , ( ≥ 2).…”

“…Using Theorem 1.1 in Section 5.1 in [9] we get that all spaces for ≥ 2 satisfy the 2 property with their canonical norm and the same result is also true for the Sobolev spaces , ( ≥ 2). We refer the reader for more examples and discussions to [9,12] and the references therein.…”

“…Assume that admits an equivalent norm ‖ ⋅ ‖ such that ‖ ⋅ ‖ (for some ≥ 2) is 2 -differentiable on \ {0}. We refer the reader to [9,12] for many examples of Banach space satisfying this assumption. So, we are ready to prove the main result.…”