Dual spaces and weak topologies

The bounded linear functionals on a normed space X form its dual X*, with operator norm given by the largest absolute evaluation on the unit ball. A norm-Cauchy sequence of functionals is pointwise Cauchy in the scalar field; its pointwise limit is linear and satisfies the inherited uniform bound. This proves X* Banach even if X is incomplete. Hahn–Banach makes the canonical evaluation map J:X→X** an isometry. Whether this map reaches every element of the bidual is the later question of reflexivity.

For sequences, the dual of ℓᵖ is ℓᑫ when 1≤p<∞ and 1/p+1/q=1, with q=∞ at p=1. The dual of c₀ is ℓ¹. Pairing means summing a bounded linear coefficient sequence against the input, with a consistent complex-linear convention rather than silently conjugating the wrong argument. The dual of ℓ∞ is larger than ℓ¹; identifying it with summable coefficients would erase important functionals. These endpoint differences are structural, not merely an inconvenient exception in notation.

The weak topology σ(X,X*) is the smallest topology making every bounded functional on X continuous. A net converges weakly exactly when all its scalar evaluations converge. The weak-star topology σ(X*,X) on the dual tests f at each fixed x in X. The weak topology on that same dual instead tests every element of X**. Since JX sits inside X**, weak convergence on X* implies weak-star convergence, but the reverse can fail. Norm convergence implies weak convergence in either setting.

A weakly convergent sequence is norm bounded by uniform boundedness. In a sequence space, boundedness plus convergence against finitely supported tests can sometimes be upgraded to all dual tests by approximating the testing coefficients in their own norm. This succeeds for the weak topology of c₀, whose dual coefficients are summable, but not for ℓ¹, whose dual coefficients can be arbitrary bounded sequences. The failure is easy to see: coordinate unit vectors have every fixed coordinate tending to zero, while the constant-one ℓ∞ functional evaluates each at one.

View ℓ∞ as the dual of ℓ¹. Let u⁽ⁿ⁾ be one in the first n coordinates and zero afterward. Against each summable sequence, the evaluations converge to its total sum, so u⁽ⁿ⁾ converges weak-star to the constant-one sequence. To test weak convergence in ℓ∞, however, one may use any element of its full dual. Extend the ordinary limit functional from the subspace c of convergent sequences to ℓ∞ by Hahn–Banach. This bounded extension vanishes on c₀ and takes value one on the constant-one sequence.

Every u⁽ⁿ⁾ belongs to c₀, so that extended functional evaluates all of them at zero. Its value at their proposed weak limit is one, ruling out weak convergence to that limit. The example needs no special translation-invariant Banach limit: an arbitrary norm-preserving extension of the ordinary limit suffices. It demonstrates why a weak-star compactness argument cannot be upgraded to a weak compactness argument just by observing bounded norms. The topology and its available scalar tests must remain explicit at every step.