Let V = R1[x], consider the map T : V −→ V given by T(p) = p(1)+p(−1)x. Suppose V has the inner product given by〈p, q〉 = p(1)q(1) + p′(1)q′(1)
What is T∗(x)? Show how to get the answer 7-4x
with inner product
Then the adjoint is found by solving
Substituting we get
Equation becomes
So that
Therefore,
Now we substitute and
So means
Therefore,
Thus,
Noting that and so that
And we get the solution as as required
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