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Theme 2 — Hilbert Spaces and Dirac Notation

Riesz representative in finite dimension

Keywords: Riesz · linear functional · inner product · antilinearity · finite dimension

Exercise 1 : Riesz representative in finite dimension

Question 1
Let E=CnE = \mathbb{C}^n be equipped with the usual inner product ⟨x,y⟩=∑k=1nxk∗yk\langle x, y \rangle = \sum_{k=1}^n x_k^* y_k. Let φ ⁣:E→C\varphi \colon E \to \mathbb{C} be the linear functional defined by φ(x)=∑k=1nakxk,(a1,...,an)∈Cn.\varphi(x) = \sum_{k=1}^n a_k x_k, \qquad (a_1, ..., a_n) \in \mathbb{C}^n.

Question 2
Determine the Riesz representative uφ∈Eu_\varphi \in E such that φ(x)=⟨uφ,x⟩\varphi(x) = \langle u_\varphi, x \rangle for every x∈Ex \in E.

Solution
We seek u=(u1,...,un)u = (u_1, ..., u_n) such that ⟨u,x⟩=∑kuk∗xk=∑kakxk\langle u, x \rangle = \sum_k u_k^* x_k = \sum_k a_k x_k for every x∈Ex \in E. Identifying the components one by one gives uk∗=aku_k^* = a_k, and hence

uφ=(a1∗,...,an∗).u_\varphi = (a_1^*, ..., a_n^*).

Question 3
Show that the map φ↦uφ\varphi \mapsto u_\varphi is antilinear.

Solution
Let φ,ψ\varphi, \psi be two linear functionals with representatives uφu_\varphi and uψu_\psi, and let λ∈C\lambda \in \mathbb{C}. The representative of λφ+ψ\lambda\varphi + \psi satisfies, for every xx:

⟨uλφ+ψ,x⟩=(λφ+ψ)(x)=λφ(x)+ψ(x)=λ⟨uφ,x⟩+⟨uψ,x⟩=⟨λ∗uφ+uψ,x⟩,\langle u_{\lambda\varphi+\psi}, x \rangle = (\lambda\varphi + \psi)(x) = \lambda\varphi(x) + \psi(x) = \lambda \langle u_\varphi, x \rangle + \langle u_\psi, x \rangle = \langle \lambda^* u_\varphi + u_\psi, x \rangle,

and therefore uλφ+ψ=λ∗uφ+uψu_{\lambda\varphi + \psi} = \lambda^* u_\varphi + u_\psi. The map φ↦uφ\varphi \mapsto u_\varphi is indeed antilinear.