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.)
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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let (an)n∈Z be a sequence of real numbers. Recall the ℓ2(ℕ), as a
?||(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.)
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