Functional Analysis

JKPSC Assistant Professor (Mathematics) · Mathematics · 49 practice questions with answers and explanations: 5 Easy, 40 Medium, 4 Hard.

Sample questions

Sample Easy question 1

Two vectors in an inner product space are orthogonal if their inner product is:

  1. Zero
  2. Infinite
  3. Equal to their norms
  4. One
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Answer: A. In ℝ² this is the usual perpendicularity.

Sample Easy question 2

In a normed space, ‖x‖ = 0 if and only if:

  1. x = 0
  2. x is a unit vector
  3. x is bounded
  4. x is real
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Answer: A. This is the definiteness axiom.

Sample Easy question 3

The triangle inequality for a norm is:

  1. ‖x + y‖ ≥ ‖x‖ + ‖y‖
  2. ‖x − y‖ ≤ ‖x‖ − ‖y‖
  3. ‖x + y‖ ≤ ‖x‖ + ‖y‖
  4. ‖x + y‖ = ‖x‖ ‖y‖
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Answer: C. Equality holds when one vector is a non-negative multiple of the other.

Sample Medium question 4

The Cauchy–Schwarz inequality in an inner product space is:

  1. ‖x + y‖ = ‖x‖ ‖y‖
  2. ⟨x, y⟩ = ‖x‖ + ‖y‖
  3. |⟨x, y⟩| ≥ ‖x‖ ‖y‖
  4. |⟨x, y⟩| ≤ ‖x‖ ‖y‖
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Answer: D. Equality holds iff x and y are linearly dependent.

Sample Medium question 5

The Riesz representation theorem says every bounded linear functional on a Hilbert space H has the form:

  1. f(x) = ⟨x, y⟩ for a unique y in H
  2. f(x) = ‖x‖
  3. f(x) = ⟨y, y⟩
  4. f(x) = x + y
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Answer: A. The norm of f equals ‖y‖.

Sample Medium question 6

The parallelogram law states that ‖x + y‖² + ‖x − y‖² =

  1. 4‖x‖ ‖y‖
  2. ‖x‖² + ‖y‖²
  3. 2‖x‖ ‖y‖
  4. 2(‖x‖² + ‖y‖²)
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Answer: D. A norm comes from an inner product iff it satisfies this law.

Sample Hard question 7

By the Riesz–Fischer theorem, L²[a, b] is:

  1. Finite-dimensional
  2. A Banach space but not a Hilbert space
  3. Complete
  4. Not complete
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Answer: C. It is a Hilbert space.

Sample Hard question 8

Which of these is NOT a Hilbert space?

  1. ℓ¹ with its usual norm
  2. ℓ²
  3. ℝⁿ with the Euclidean norm
  4. L²[0, 1]
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Answer: A. The ℓ¹ norm does not satisfy the parallelogram law.

Sample Hard question 9

The dual space of ℓ¹ is:

  1. ℓ^∞
  2. ℓ²
  3. ℓ¹
  4. c₀
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Answer: A. The dual of c₀ is ℓ¹.

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