Real Analysis by Folland Gerald B.;

Author:Folland, Gerald B.;
Language: eng
Format: epub
ISBN: 1655934
Publisher: Wiley
Published: 2013-06-09T16:00:00+00:00

which tends to zero since ||yn|| → ||y||.

5.22 The Parallelogram Law. For all x, y

(“The sum of the squares of the diagonals of a parallelogram is the sum of the squares of the four sides.”)

Proof. Add the two formulas ||x ± y||2 = ||x||2 ± 2 Rex, y + ||y||2.

If x, y , we say that x is orthogonal to y and write x ⊥ y if x, y = 0. If E ⊂ , we define

It is immediate from Proposition 5.21 and the linearity of the inner product in its first argument that E⊥ is a closed subspace of .

5.23 The Pythagorean Theorem. If x1,…, xn and xj ⊥ xk for j ≠ k,

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