The final section of Chapter 3 deals with continuous mappings.
Thus it is an inner product (a weighted Euclidean inner product). The induced norm is (|x| = \sqrt\sum w_k x_k^2). kreyszig functional analysis solutions chapter 3
‖x+y‖2=‖x‖2+‖y‖2the norm of x plus y end-norm squared equals the norm of x end-norm squared plus the norm of y end-norm squared Solutions often involve extending this to mutually orthogonal vectors using induction to show 3. Completeness of Subspaces (Section 3.2) The final section of Chapter 3 deals with
‖x+y‖2+‖x−y‖2=2⟨x,x⟩+2⟨y,y⟩the norm of x plus y end-norm squared plus the norm of x minus y end-norm squared equals 2 open angle bracket x comma x close angle bracket plus 2 open angle bracket y comma y close angle bracket kreyszig functional analysis solutions chapter 3
(M = (x_n) : x_2k=0 \ \forall k ) (sequences with zeros at even indices).
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