Solvability of main and adjoint equations in non-linear problems

Let X and Y be two spaces in a sequence {H^α } of real Hilbert spaces introduced above, among which only H ₀ ≡ H ⁰ is identified with its dual space, that is, constitutes the basic space. We also assume that the space (H^α )^* dual to H^α , 0 ≤ α ≤ 1, is identified with H ^–α. When considering the spaces X and Y, we denote the scalar products, norms and duality relations by ( f ,   g ) X ,   ‖ f ‖ X   =   ( f ,   f ) X 1 / 2   〈 f ,   g * 〉 X   ≡   ( f ,   g * ) H 0 ,   f ,   g   ∈   X ,   g *   ∈   X * , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136707/b8efdbc7-0f66-4bda-8448-b1ca27e88da4/content/eq388.tif"/> ( f ,   g ) Y ,   ‖ f ‖ Y   =   ( f ,   f ) Y 1 / 2   〈 f ,   g * 〉 Y   ≡   ( f ,   g * ) H 0 ,   f ,   g   ∈   Y ,   g *   ∈   Y * , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136707/b8efdbc7-0f66-4bda-8448-b1ca27e88da4/content/eq389.tif"/>