The Riesz Representation Theorem states that every bounded linear functional on a Hilbert space H has the form f(x) = ⟨x, y⟩ for a unique y ∈ H. A student concludes: 'So H* and H are the same space.' What is the correct interpretation?
AThe student is correct — H* and H are identical as sets and as normed spaces
BH* is isometrically isomorphic to H, meaning there is a norm-preserving bijection, but H* is formally a distinct space whose elements are functionals, not vectors
CH* is a proper subset of H for most Hilbert spaces
DThis isomorphism holds only for finite-dimensional Hilbert spaces
The Riesz theorem gives a natural isometric isomorphism between H and H*, but they are not literally the same object. H consists of vectors; H* consists of bounded linear functionals. The isomorphism says every functional can be 'identified with' a vector, and the identification preserves norms. The student's language is imprecise but points at the right idea: Hilbert spaces are 'self-dual' in this sense. Contrast this with ℓ¹, whose dual is ℓ^∞, which is a strictly larger space — very different from its original.
Question 2 Multiple Choice
What is the dual space (ℓ²)*?
Aℓ¹ — the space of absolutely summable sequences
Bℓ^∞ — the space of bounded sequences
Cℓ² itself — by the Riesz Representation Theorem for Hilbert spaces
Dc₀ — sequences converging to zero
ℓ² is a Hilbert space (with inner product ⟨x,y⟩ = Σ xₙȳₙ), so the Riesz Representation Theorem applies: every bounded functional on ℓ² has the form f(x) = Σ aₙxₙ for some (aₙ) ∈ ℓ², giving (ℓ²)* ≅ ℓ². The general Hölder duality rule (ℓᵖ)* ≅ ℓ^q where 1/p+1/q=1 gives 1/2+1/2=1, consistent with the self-dual answer. ℓ¹ is the dual of c₀; ℓ^∞ is the dual of ℓ¹. These are distinct cases that confirm ℓ² is exceptional.
Question 3 True / False
A sequence (xₙ) converges weakly to x in a normed space X if and only if f(xₙ) → f(x) for every bounded linear functional f in X*.
TTrue
FFalse
Answer: True
This is the definition of weak convergence. The weak topology on X is precisely the coarsest topology that makes every functional in X* continuous. Weak convergence is weaker than norm convergence: if ‖xₙ − x‖ → 0 (norm convergence) then f(xₙ) → f(x) for all f by continuity, but the converse fails in infinite-dimensional spaces. A standard example is the standard basis (eₙ) in ℓ², which converges weakly to 0 but has ‖eₙ‖ = 1 for all n.
Question 4 True / False
If a sequence converges weakly in an infinite-dimensional Banach space, it is expected to also converge in norm.
TTrue
FFalse
Answer: False
Weak convergence does not imply norm convergence in infinite-dimensional spaces. The standard basis vectors (eₙ) in ℓ² converge weakly to 0 — for any f ∈ (ℓ²)*, represented by y ∈ ℓ², f(eₙ) = yₙ → 0 because y ∈ ℓ² means its terms go to zero. But ‖eₙ − 0‖ = ‖eₙ‖ = 1 for all n. This failure is one reason weak topology is useful: it allows sequences to 'converge' in a useful sense without the strong requirement of norm convergence, enabling compactness arguments unavailable in the norm topology.
Question 5 Short Answer
Explain intuitively why the dual space X* can be thought of as a tool for 'testing' or 'measuring' elements of X, and why this perspective is useful in functional analysis.
Think about your answer, then reveal below.
Model answer: Each functional f ∈ X* extracts a single real number from any vector x ∈ X. The collection of all such measurements — the entire dual space — gives a complete picture of x: if f(x) = f(y) for all f ∈ X*, then x = y (by Hahn-Banach). Studying X through X* is like studying a physical object by all possible measurements you could take of it. This perspective is useful because dual space methods convert problems about vectors (which may be abstract functions or sequences) into problems about numbers, enabling existence proofs, compactness arguments, and optimization via duality.
The weak topology formalizes this: it is the topology on X induced by all the measurements in X*. Two points are 'close' in the weak topology if all functionals give nearby values on them. This topology is coarser than the norm topology, which is why weak compactness is easier to achieve — a powerful tool in proving existence theorems in PDEs, calculus of variations, and optimization.