... (0, 2 ) = u ∈ L2 (0, 2 ) : {um } , {mum } ∈ l2 Hper and introduce the inner product + m2 um v−m (u, v)1 ,2 = (2 ) m∈Z which makes Hper (0, 2 ) into a Hilbert space Since {mum } ∈ lC2 , (0, 2 ) ... (λ1 u1 + 2 u2 , v) = λ1 a (u1 , v) + 2 a (u2 , v) = λ1 (A [u1 ] ,v) + 2 (A [u2 ] ,v) = (λ1 A [u1 ] + 2 A [u2 ] ,v) whence A [λ1 u1 + 2 u2 ] = λ1 A [u1 ] + 2 A [u2 ] Thus A is linear and ... space is L2 (Ω), Ω ⊆ Rn In particular, the set of functions cos x sin x cos 2x sin 2x cos mx sin mx √ , √ , √ , √ , √ , , √ , √ , π π π π π π 2 constitutes an orthonormal basis in L2 (0, 2 ) (see...