A Friendly Approach to Functional Analysis by Amol Sasane

Author:Amol Sasane
Language: eng
Format: epub
ISBN: 9781786343369
Publisher: World Scientific Publishing Co. Pte. Ltd.
Published: 2017-01-06T05:00:00+00:00

(2)If e1 := (1, 0, ···) and e2 = (0, 1, 0, ···) in ℓ2, then

Exercise 4.16. Let Y be a subspace of an inner product space X.

Show that Y ∩ Y⊥ = {0}.

Exercise 4.17. Let Y be a subspace of an inner product space X.

(1)Prove that Y ⊂ (Y⊥)⊥.

(2)Show that if Y ⊂ Z, where Z is another subspace of X, then Z⊥ ⊂ Y⊥.

(3)Prove that Y⊥ = ⊥, where denotes the closure of Y.

(4)Show that if Y is dense in X, then Y⊥ = {0}.

(5)Let Yeven be the subspace of ℓ2 of all sequences whose oddly indexed terms are zeros. Describe Show that .

(6)What is in ℓ2? Show that .

(Thus for a subspace Y, it can happen that Y ≠ Y⊥⊥.

We’ll see later that for a closed subspace Y in a Hilbert space H, Y = Y⊥⊥.

But for general subspaces Y, we can only say that Y⊥⊥ = .)

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