Question

Asked Dec 29, 2019

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Complete the proof of Theorem 6.2.

Theorem 6.2. Let V be an inner product space over F. Then for all x, y ∈V and c ∈F, the following statements are true.

(a) ||cx||= |c|.||x||.

(b) ||x||= 0 if and only if x = 0 . In any case, ||x||≥0.

(c) (Cauchy–Schwarz Inequality) |<x, y>| ≤<||x||·||y||.

(d) (Triangle Inequality) ||x + y||≤|x||+ ||y||.

Step 1

Let V be an inner product space over F. Then for all x ,y € V and c € F,

For part (a)

It is required to prove that: ||cx|| = |c|. ||x||

Consider

Step 2

For part (b) it is required to prove that ||x|| = 0 if and only if x = 0.

Consider,

Step 3

For part (c) it is required to prove that: |<x...

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