{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Lecture02

# Lecture02 - Lecture 2 Exercise If dim V> 1 then ‘the’...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Lecture 2 Exercise : If dim V > 1 then ‘the’ ‘ P norm arises from an I.P. precisely when p = 2. Proof. ( ⇒ ) Since dim V > 1, there are at least two vectors v 1 , v 2 in the basis used to define the norm. If the ‘ p nrom comes from an I.P. then the parallelogram law must hold: k v 1 + v 2 k 2 p + k v 1- v 2 k 2 p = 2( k v 1 k 2 p + k v 2 k 2 p ) ((1 p + 1 p ) 1 p ) 2 ((1 p + 1 p ) 1 p ) 2 1 2 1 2 So 2 · 2 1 p = 2 · 2 and hence 2 p = 1 or p = 2. ( ⇐ ) This was seen before. Exercise : If V is infinite-dimensional, then it is not complete for the ‘ ∞ norm associated to a basis. Proof. Since the dimension of V is infinite, we can find a countable (sub)collection ( v m ) ∞ m =1 of basis vectors used to define the norm. Define a new sequence ( w m ) ∞ m =1 by setting w m = ∑ m k =1 1 k v k . Then for n > m we have: k w n- w m k ∞ = k 1 n v n + 1 n- 1 v n- 1 + ··· + 1 m + 1 v m +1 k ∞ Thus d ∞ ( w n ,w m ) = max { 1 n , 1 n- 1 ,..., 1 m +1 } = 1 m +1 ≤ if n > m > 1 and so ( w m ) ∞ m =1 is Cauchy. Informally, if ( w m ) ∞ m =1 converges then w m → ∞ X k =1 1 k v k / ∈ V (since every vector is a finite linear combination of the...
View Full Document

{[ snackBarMessage ]}

### Page1 / 4

Lecture02 - Lecture 2 Exercise If dim V> 1 then ‘the’...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online