{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Macroeconomics Exam Review 114

# Macroeconomics Exam Review 114 - c 2001 Michael Carter All...

This preview shows page 1. Sign up to view the full content.

We need to show that the sequence ( 𝑝 𝑡 ) 𝑙 1 . For any 𝑁 , consider the sequence x 𝑡 = ( 𝑥 1 , 𝑥 2 , . . . , 𝑥 𝑡 , 0 , 0 , . . . ) where 𝑥 𝑡 = 0 𝑝 𝑡 = 0 or 𝑛 𝑁 𝑝 𝑡 𝑝 𝑡 otherwise Then ( x 𝑡 ) 𝑐 0 , x 𝑡 = 1 and 𝑓 ( x 𝑡 ) = 𝑡 𝑛 =1 𝑝 𝑡 𝑥 𝑡 = 𝑡 𝑛 =1 𝑝 𝑡 Since 𝑓 𝑐 0 , 𝑓 is bounded and therefore 𝑓 ( x 𝑡 ) ≤ ∥ 𝑓 ∥ ∥ x 𝑡 = 𝑓 < and therefore 𝑡 𝑛 =1 𝑝 𝑡 < for every 𝑁 = 1 , 2 , . . . Consequently 𝑛 =1 𝑝 𝑡 = sup 𝑁 𝑡 𝑛 =1 𝑝 𝑡 ∣ ≤ ∥ 𝑓 < We conclude that ( 𝑝 𝑡 ) 𝑙 1 and therefore 𝑐 0 = 𝑙 1 3. Similarly, every sequence ( 𝑝 𝑡 ) 𝑙 defines a linear functional 𝑓 on 𝑙 1 given by 𝑓 ( x ) = 𝑛 =1 𝑝 𝑡 𝑥 𝑡 for every x = ( 𝑥 𝑡 ) 𝑙 1 . Moreover 𝑓 is bounded since 𝑓 ( x ) ∣ ≤ 𝑛 =1 𝑝 𝑡 ∣ ∣ 𝑥 𝑡 ∣ ≤ ∥ ( 𝑝 𝑡 ) 𝑛 =1 𝑥 𝑡 < for every x = ( 𝑥 𝑡 ) 𝑙 1 Again, given any linear functional
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Ask a homework question - tutors are online