Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 326 / 1 / 3 / a

K : U \to V

consists of positive singular values

σ_{j}

and orthonormal systems

u_{j} \in U

v_{j} \in V

satisfying

K u_{j} = σ_{j} v_{j}, K^{*} v_{j} = σ_{j} u_{j} .

They span

N (K)^{⊥}

and

\overline{R (K)}

K u = j \sum σ_{j} ⟨ u, u_{j} ⟩ v_{j} .

For an infinite-rank compact operator, the singular values tend to zero; for finite rank the sum is finite. Vectors in

N (K)

contribute nothing. This convention labels the input singular vectors by

u_{j}

and the output ones by

v_{j}

For

(K u)_{j} = u_{j} / j

, truncating after

n

1/ (n + 1)

, proving compactness. Its SVD is

u_{j} = v_{j} = e_{j}, σ_{j} = 1/ j .

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