Convex functional analysis

By A. Kurdila, Michael Zabarankin. 2005

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On the optimal control of variational–hemivariational inequalities (2019, background): “rcive, i.e., J(v) → ∞ as kvkX → ∞. Then, there exists at least an element u such that u ∈ K, J(u) ≤ J(v) ∀v ∈ K. (2.1) The proof of Theorem 5 can be found in many books and survey, see, for instance, [11, 24]. We now introduce the variational-hemivariational inequality we are interested in and state”
Distribution-on-Distribution Regression via Optimal Transport Maps (2021, methodology): “To show the existence, we use a variant of the Weierstrass theorem, namely Kurdila and Zabarankin [8, Thm 7.3.6], stated for convenience as Theorem 6.1 in the Appendix.”
Multiple Operator-valued Kernel Learning (2012, background): “Before deriving the dual of this problem, it can be shown by means of the generalized Weierstrass theorem [17] that this problem admits a solution.”
Operator-valued Kernels for Learning from Functional Response Data (2015, methodology): “…problem F̃λ = arg min F∈F n∑ i=1 ‖yi − F (xi)‖2Y + λ‖F‖2F has the following form F̃λ(·) = n∑ i=1 K(xi, ·)ui, where ui ∈ Y. Proof : We use the Frechet derivative which is the strongest notion of derivative in a normed linear space; see, for example, Chapter 4 of Kurdila and Zabarankin (2005).”