It would be pedagogically useful to be given an a

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to use at this stage of knowledge. It would be pedagogically useful to be given an a priori argument for why one would consider this line of attack (assuming knowledge which a student can be assumed to know up to this point) at the outset . Of course, one can ask the question as to which value of α gives the tightest bound, and the answer to this question shows that the C-S inequality is optimal in this sense. However, as mentioned, many textbooks follow the proof we have detailed given above. Note that by asking for the specific value of α for which the always nonnegative quantity x αy 2 is minimized, Moon is effectively asking for the optimal projection of x onto the one-dimensional subspace spanned by y in the least-squares sense. This optimal choice of α happens to be equal to the value we determined above. 2. Moon 2.1-4. The proof for both cases is essentially the same as R 1 is just a specific instance of the general R n case. We have (take α = 1 in equation (1) above), x + y 2 = x 2 + 2 Re x, y + y 2 x 2 + 2 | ⟨ x, y ⟩ | + y 2 x 2 + 2 x ∥ ∥ y + y 2 ( x + y ) 2 . 1
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Taking the square root of both sides yields the triangle inequality. 3. Moon 2.12-57 and 2.12-58. All the desired properties of vector addition and scalar multiplication are inherited from the parent space.
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