(c) For an orthonormal sequence {u} in a Hilbert space H, prove the equivalence of the conditions (i) xlu for all n implies x = 0, (ii) x = Σu for all x € H.

Q: Consider the sequence aj = 12 a2 = /12 + V12 as = /12+ V12+ Vī2 12 + V12 12 + a4 = 12 + 1/12 + 12 +…

A: a) Given sequence is: a1=12a2=12+12a3=12+12+12...

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A: Every subsequence has a pattern.  According to guidelince we solve only first question.

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A:

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Q: Q: Use the Squeeze theorem to estabich cennergence of the gi , ren Sequence Conr Inx X(X+2)

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A: Proof

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A:

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A: Solution is in step 2

Q: 14. Prove that every Cauchy sequence is bounded (Theorem 1.4)

A: To prove: Every Cauchy sequence is bounded.

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Q: Prove from first principles that any Cauchy sequence (an) is bounded.

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A: 3.

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A: NOTE:- I HAVE REQUIRED QUESTION A TO SOLVE QUESTION C AND D. THEREFORE I AM SOLVING ONLY QUESTION B.

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