Let (an)n∈Z be a sequence of real numbers. Recall the ℓ2(ℕ), as a vector space over R, has the norm given by ?||(an)||ℓ2(ℕ) = (Σ∞n=1 |an|2)1/2 =  ?limN→∞(ΣNn=1 ?|an|2)1/2 . We will often denote this by ||an||ℓ2 to have a cleaner notation. (a) Let (an) and (bn) ∈ ℓ2(ℕ). Prove the Cauchy-Schwarz inequality, |⟨(an), (bn)⟩| = ||an||ℓ2 ||bn||ℓ2 , which in this inner product space, takes the form: |Σn=ℕ anbn| = (Σn=ℕ |an|2)1/2 (Σn=ℕ |bn|2)1/2 (b)  Prove that || · ||ℓ2 satisfies the triangle inequality. (Once we have this fact, it’s now easy to see that ℓ2 is a normed linear space.)

Advanced Engineering Mathematics
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ISBN:9780470458365
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Chapter2: Second-order Linear Odes
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Let (an)n∈Z be a sequence of real numbers. Recall the 2(ℕ), as a vector space over R, has the norm given by

?||(an)||2(ℕ) = (Σn=1 |an|2)1/2  ?limN→∞Nn=1 ?|an|2)1/2 .

We will often denote this by ||an||2 to have a cleaner notation.

(a) Let (an) and (bn) ∈ 2(ℕ). Prove the Cauchy-Schwarz inequality,

|⟨(an), (bn)⟩| = ||an||2 ||bn||2 , which in this inner product space, takes the form:

n=ℕ anbn| = (Σn=ℕ |an|2)1/2n=ℕ |bn|2)1/2

(b)  Prove that || · ||2 satisfies the triangle inequality. (Once we have this fact, it’s now easy to see that 2 is a normed linear space.)

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