Completion Notes

# We must prove that it has a limit x h n as n each xn

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Unformatted text preview: −→ 0 so that addition and scalar multiplication are well–deﬁnd. Lemma 6 H with the operations of Lemma 5 is an inner product space. Proof: There are nine vector space axioms and three inner product axioms to be checked. The proofs are essentially trivial and very similar. I’ll just verify the ﬁrst vector space axiom and ﬁrst half of the ﬁrst inner product axiom. Let {xn }n∈IN , {yn }n∈IN ∈ H and α ∈ C. {xn }n∈IN + {yn }n∈IN = {xn + yn }n∈IN = {yn + xn }n∈IN = {yn }n∈IN + {xn }n∈IN {xn }n∈IN , α {yn }n∈IN H = {xn }n∈IN , {αyn }n∈IN H = α {xn }n∈IN , {yn }n∈IN = lim xn , αyn n→∞ V = α lim xn , yn n→∞ V H Lemma 7 H is complete Proof: Let {X(n) ∈ H}n∈IN be a Cauchy sequence. We must prove that it has a limit, X ∈ H, (n) as n → ∞. Each X(n) is an equivalence class of Cauchy sequences in V . Say X(n) = x . ∈IN (n) with chosen larger and larger We shall guess X = xn n∈IN by choosing each xn to be a x as n increases. Here we go: (1) (1) (1) (1) x Cauchy ⇒ ∃ 1 ∈ IN s.t. ≥...
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