 Solution

ID: past-exam-of-the-algebra-analysis-and-differential-equations-2-course-of-the-university-of-oxford/2011/algebra/2/a/ii/solution

The adjoint operator is characterized by

⟨ T u, v ⟩ = ⟨ u, T^{*} v ⟩ for all u, v \in V .

For each fixed

v

, the map

u \mapsto ⟨ T u, v ⟩

is a linear functional. Part (i) gives a representing vector, which we define to be

T^{*} v

. Hence such a map exists on every vector of

V

. Part (b) proves that this constructed map is linear.

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