Questions: Dual Spaces and Bounded Linear Functionals

Question 1

Let X be a normed vector space that is not complete (not a Banach space). Which of the following best describes its dual space X*?

• A. X* is also incomplete, since it inherits the incompleteness of X
• B. X* is always a Banach space, regardless of whether X is complete
• C. X* is empty if X is not complete, since bounded functionals require completeness
• D. X* has the same topology as X but a different norm

Answer: (B)

The dual space X* is automatically a Banach space — complete under the operator norm — even when X itself is incomplete. This follows because bounded linear maps into a complete space inherit completeness: the scalar field ℝ is complete, and a Cauchy sequence of bounded functionals {φₙ} converges pointwise (since {φₙ(x)} is Cauchy in ℝ for each x), defining a limit functional that is bounded and linear. The argument makes no use of completeness of X. This is one of the most useful properties of dual spaces: X* is always a Banach space regardless of what X is.

Question 2

The statement 'X* separates points of X' means which of the following?

• A. Every individual bounded functional on X maps distinct elements to distinct real numbers
• B. If φ(x) = φ(y) for every φ ∈ X*, then x = y
• C. X* contains at least one functional that is injective on all of X
• D. Bounded functionals can distinguish vectors only within finite-dimensional subspaces

Answer: (B)

The separation property says: if every functional in X* assigns the same value to x and y, then x = y — equivalently, distinct vectors can be distinguished by some functional in X*. This is a property of the *collective* dual space, not any individual functional (option A is too strong — no single φ need be injective on all of X). The separation property is guaranteed by the Hahn-Banach theorem and means no information about X is 'hidden' from its dual: the collection of measuring devices in X* sees everything.

Question 3

The dual space X* of any normed vector space X is automatically a Banach space (complete under the operator norm), even if X itself is not a Banach space.

• T. True
• F. False

Answer: True

True. Completeness of X* follows from the completeness of the scalar field ℝ, not from X. If {φₙ} is Cauchy in X*, then for each fixed x ∈ X, the sequence {φₙ(x)} is Cauchy in ℝ (since |φₙ(x) − φₘ(x)| ≤ ‖φₙ − φₘ‖·‖x‖), hence converges. This defines a limit functional φ, which one verifies is bounded and linear, with ‖φₙ − φ‖ → 0. The completeness of X is never invoked. This automatic completeness is a principal reason dual spaces are preferred in analysis.

Question 4

The dual of the Banach space Lᵖ(μ) (for 1 < p < ∞) consists of most bounded linear functionals of the form f ↦ ∫fg dμ where g ∈ Lᵖ(μ) — that is, g lives in the same Lᵖ space.

• T. True
• F. False

Answer: False

False. The dual of Lᵖ(μ) is Lᵍ(μ) where the exponents satisfy 1/p + 1/q = 1 — the Hölder conjugate, not Lᵖ itself. For example, the dual of L³(μ) is L^(3/2)(μ), not L³(μ). The Riesz representation theorem states every φ ∈ (Lᵖ)* has the form φ(f) = ∫fg dμ for a unique g ∈ Lᵍ, and ‖φ‖ = ‖g‖_q. The Hölder conjugate relationship 1/p + 1/q = 1 is precisely what makes ∫fg dμ well-defined and bounded via Hölder's inequality. (The special case p = q = 2 — the dual of L² is L² — is where the misconception likely originates.)

Question 5

Explain intuitively what a bounded linear functional 'does' geometrically to a normed space X, and how this relates to the idea that X* encodes information about X.

Model answer: A bounded linear functional φ: X → ℝ partitions X into parallel level hyperplanes {x : φ(x) = c}. It is a 'projection' of X onto the real line, picking out one linear 'direction.' The collection of all such projections in X* collectively encodes the full geometry of X: two vectors that every functional maps to the same value must be identical (the separation property). The norm of φ measures how finely it distinguishes nearby vectors. For example, on L²([0,1]), the functional φ_g(f) = ∫f(x)g(x)dx measures how much f resembles g in a precise inner-product sense.

This geometric picture becomes precise through the Hahn-Banach theorem, which guarantees enough functionals exist to separate any two points and to 'touch' every point on the unit sphere. Duality also encodes convexity: closed convex sets in X are intersections of half-spaces defined by functionals. This makes dual spaces central to optimization (Lagrangian duality), PDEs (weak formulations treat solutions as elements of dual spaces), and the representation theory of function spaces.