T : V \to V

T^{*} : V \to V

satisfying

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

This identity transfers

T

from the first input of the inner product to the second.

ii

Solution

R

and

S

both satisfy the defining identity of an adjoint operator, then

⟨ u, (R - S) w ⟩ = ⟨ T u, w ⟩ - ⟨ T u, w ⟩ = 0

for every

u, w \in V

. Choosing

u = (R - S) w

gives

∥ (R - S) w ∥^{2} = 0

(R - S) w = 0

for every

w

, proving

R = S .

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