 Solution

ID: past-exam-of-the-algebra-analysis-and-differential-equations-1-course-of-the-university-of-oxford/2013/algebra/2/a/solution

T^{*}

is characterized by

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

This identity works with either convention for which argument of the complex inner product is linear, provided the convention is used consistently.

For uniqueness, subtract the two given identities to get

⟨ v, (S - T^{*}) w ⟩ = 0

for every

v, w

. Fix

w

and choose

v = (S - T^{*}) w

∥ (S - T^{*}) w ∥^{2} = 0,

(S - T^{*}) w = 0

w

was arbitrary,

S = T^{*}

. The decisive fact is that a vector orthogonal to every vector, including itself, must be zero.

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