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**Additional resources for Differential Equations in Banach Spaces: Proceedings of a Conference held in Bologna, July 2–5, 1985**

**Example text**

Proof: We prove this result by induction on the dimension. For a zero-dimensional Hilbert space there is nothing to show. So let V be a pre-Hilbert space of dimension k + 1 and assume that the claim has been proven for all spaces of dimension k. Let v ∈ V be a nonzero vector of norm 1. , the space of all u ∈ V with u, v = 0. 10) and the dimension of U is k, so this space is complete by the induction hypothesis. Let (vn ) be a Cauchy sequence in V ; then for each natural number n, vn = λn v + un , where λn is a complex number and un ∈ U .

S∈S Show that |f (s)| ||f ||1 = s∈S deﬁnes a norm on l1 (S). 8 For which s ∈ C does the function f (n) = n−s belong to 2 (N)? For which does it belong to l1 (N)? 9 For T > 0 let C([−T, T ]) denote the space of all continuous functions f : [−T, T ] → C. Show that the prescription T f, g f (x)g(x)dx = −T for f, g ∈ C([−T, T ]) deﬁnes an inner product on this space. 10 Let V be a ﬁnite-dimensional pre-Hilbert space and let W ⊂ V be a subspace. , U is the space of all u ∈ V such that u, w = 0 for every w ∈ W .

We then have that ϕ(t) = v + tw, v + tw = ||v + tw||2 ≥ 0. Note that v, w + w, v = 2Re v, w . The real-valued function ϕ(t) is a quadratic polynomial with positive leading coeﬃcient. , at the point t0 = −Re v, w / w 2 . Evaluating at t0 , we see that 0 ≤ ϕ(t0 ) = ||v||2 + (Re v, w )2 (Re v, w )2 − 2 , ||w||2 ||w||2 which implies (Re v, w )2 ≤ ||v||2 ||w||2 . Replacing v by eiθ v for a suitable real number θ establishes the initial claim. We now show that this result implies the triangle inequality.