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.)

Q: 1. Suppose that x is a non-zero vector in a normed vector space V over F, and that a > 0. Find a…

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Q: 8.0. If x-(x,, X, ...,%)«R', define [xL by Prove that xJkL is a norm on R'. N I|} dns - x]

A: We will solve the first question highlighted in yellow. If you want the second question to be…

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A: Please find the answer in next step

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A: The solution is given as

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A: Given that L :  V→W is a linear transformation and that dim(V)=n.

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Q: 6. Consider a normed vector space S = (R,|- |). Show that, for all r, y e S, |z - y| 2 | – l9|-

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A: The given sequence is: xnt=4cosn3t5+cosnt a=-5π, b=-4π

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Q: Find the boundary of the sequence xn in the normalized vector space C [a, b], if it exists. 4 cos/3…

A: Solution: Here the sequence is xnt=4cosn3t5+cosnt and a=-5π,  b=-4π

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A: 1

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Q: This result is known as Bessel's inequality. Problem 5. Let V be a finite-dimensional inner product…

A: Please check further steps for solution.!

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A: Given  V is the set of sequences of real numbers. So we can write V as:

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Q: 10. If A is 3 x 3 with rank A = 2, show that the dimension of the null space of A is 1.

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