Let be an -dimensional vector space over with a chosen basis

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Linear algebra and vector spaces

Conditions of Validity, Ch Vector spaces, linear maps and the dual: the Riesz theorem

Problem

Let VV be an nn-dimensional vector space over C\mathbb C with a chosen basis {ei}i=1n\{\mathbf e_i\}_{i=1}^n, and let {ej′}j=1n\{\mathbf e'_j\}_{j=1}^n be a second basis related to the first by ej′=Sij ei(sum over i),\mathbf e'_j = S^i{}_j\,\mathbf e_i \qquad (\text{sum over } i), where S=(Sij)\mathsf S=(S^i{}_j) is an invertible matrix. ((a)) A vector v∈V\mathbf v\in V has components viv^i in {ei}\{\mathbf e_i\} and v′jv'^j in {ej′}\{\mathbf e'_j\}. Starting from v=viei=v′jej′\mathbf v=v^i\mathbf e_i=v'^j\mathbf e'_j, derive the relation between the column vectors vv and v′v'. Then, for a linear functional ω∈V∗\omega\in V^* with ωi=ω(ei)\omega_i=\omega(\mathbf e_i) and ωj′=ω(ej′)\omega'_j=\omega(\mathbf e'_j), derive the relation between ω′\omega' and ω\omega, and show that the pairing ωivi\omega_iv^i has the same value as ωj′v′j\omega'_jv'^j. ((b)) Suppose VV now carries an inner product ⟨ ⋅ , ⋅ ⟩\langle\,\cdot\,,\,\cdot\,\rangle, conjugate-linear in the first argument, with gij=⟨ei,ej⟩g_{ij}=\langle\mathbf e_i,\mathbf e_j\rangle forming an invertible (Gram) matrix. Prove the Riesz representation theorem: for every ω∈V∗\omega\in V^* there is exactly one u∈V\mathbf u\in V such that ω(v)=⟨u,v⟩\omega(\mathbf v)=\langle\mathbf u,\mathbf v\rangle for every v∈V\mathbf v\in V.

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