Continuous Dual of Normed Spaces: Explained with Examples

The continuous dual of a normed space X, written X*, is the space of all bounded linear functionals on X: linear maps f : X -> scalars (real or complex) satisfying |f(x)| <= M ||x|| for some constant M. With the norm ||f|| = sup{|f(x)| : ||x|| <= 1}, the dual is always a Banach space, even when X itself is not complete.

The dual matters because functionals are the "measurements" you can perform on vectors: integration against a fixed function, evaluation of a Fourier coefficient, or the expected value of a random variable. Duality turns questions about a space into questions about its functionals, and the Hahn-Banach theorem guarantees there are always enough functionals to separate points, which makes the dual rich enough to recover the geometry of X.

Key Concepts

• Linear functional: a linear map from X to the scalar field; bounded means continuous, with |f(x)| <= ||f|| ||x||.
• Dual norm: ||f|| = sup_{||x|| <= 1} |f(x)|, the maximum response of the functional to unit inputs.
• Canonical examples: on R^n, every functional is f(x) = a . x for a fixed vector a; on l^p (1 <= p < infinity) the dual is l^q with 1/p + 1/q = 1.
• Riesz representation: on a Hilbert space H, every bounded functional has the form f(x) = ⟨x, y⟩ for a unique y, so H* is identified with H itself.
• Reflexivity: X is reflexive when the natural map into its bidual X** is onto; Hilbert spaces and l^p (1 < p < infinity) are reflexive, l^1 is not.
• Weak topology: the coarsest topology making every functional in X* continuous; bounded sequences may converge weakly without converging in norm.

Formulas

• Dual norm on R^n: for f(x) = a_1 x_1 + … + a_n x_n with the Euclidean norm, ||f|| = sqrt(a_1^2 + … + a_n^2) = ||a||_2, by Cauchy-Schwarz with equality along a.
• l^p duality: for 1 <= p < infinity and 1/p + 1/q = 1, every functional on l^p is f(x) = sum x_n y_n for a unique y in l^q, with ||f|| = ||y||_q.
• Holder estimate: |sum x_n y_n| <= ||x||_p ||y||_q justifies the pairing and the norm equality.
• Riesz map: f_y(x) = ⟨x, y⟩ satisfies ||f_y|| = ||y||, giving an isometric anti-linear identification H* = H.
• Hahn-Banach consequence: for every nonzero x there is f in X* with ||f|| = 1 and f(x) = ||x||, so ||x|| = sup_{||f|| <= 1} |f(x)|.

Worked Example

Compute the dual norm of the functional f(x, y) = 3x - 2y on R^2 with the Euclidean norm. For any vector v = (x, y), Cauchy-Schwarz gives |f(v)| = |(3, -2) . (x, y)| <= ||(3,-2)||_2 ||v||_2 = sqrt(9 + 4) ||v||_2 = sqrt(13) ||v||_2, so ||f|| <= sqrt(13), about 3.606. Equality in Cauchy-Schwarz holds when v is parallel to (3, -2); take the unit vector u = (3/sqrt(13), -2/sqrt(13)). Then f(u) = 9/sqrt(13) + 4/sqrt(13) = 13/sqrt(13) = sqrt(13). Hence the supremum over unit vectors is attained and ||f|| = sqrt(13) exactly. This is the general pattern: on Euclidean space the dual norm of a linear functional is the Euclidean norm of its coefficient vector.

Common Misconceptions

• "The dual of l^1 is l^infinity, so duality is symmetric." (l^1)* is indeed l^infinity, but (l^infinity)* is strictly larger than l^1; duality need not be symmetric.
• "Every linear functional is bounded." In infinite dimensions there are discontinuous linear functionals (they need the axiom of choice to construct); the dual contains only the bounded ones.
• "X* determines X completely." Non-reflexive spaces like l^1 are strictly smaller than their biduals, so the dual sees the space only up to this gap.

Real-World Applications

• Optimization: Lagrange multipliers are functionals in the dual space; duality theory in convex optimization pairs primal and dual problems.
• Partial differential equations: weak formulations test the equation against functionals, living naturally in dual Sobolev spaces.
• Probability: expectations are functionals on spaces of random variables; the dual viewpoint underlies risk measures.
• Signal processing: sampling a signal at points or against waveforms applies functionals; reconstruction is duality in action.

The continuous dual collects all bounded linear measurements on a space, with the dual norm measuring their sensitivity. On Hilbert space every functional is an inner product, on l^p the dual is l^q, and Hahn-Banach guarantees the dual is always large enough to see every vector. Think of X* as the space of all questions you can linearly ask about X.