sol1 - cep 2008/09 MT3802 Numerical Analysis SOLUTIONS...

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Unformatted text preview: cep 2008/09 MT3802 - Numerical Analysis SOLUTIONS - Tutorial Sheet 1 1. Show that the following satisfy the requirements of a norm specified on a Vector Space: (i) k f k = Z b a | f ( x ) | dx where f ( x ) ∈ C [ a, b ] ; • Proof that is satisfies Rule 1 (1.1) f 6 = & continuous ⇒ | f ( x ) | > somewhere in [ a, b ] ⇒ Z b a | f ( x ) | dx > ⇒ k f k > where f 6 = • Proof that is satisfies Rule 2 (1.2) k α f k = Z b a | αf ( x ) | dx = Z b a | α || f ( x ) | dx = | α | Z b a | f ( x ) | dx = | α |k f k • Proof that is satisfies Rule 3 (1.3) k f + g k = Z b a | f ( x ) + g ( x ) | dx & | f ( x ) + g ( x ) | < | f ( x ) | + | g ( x ) | ⇒k f + g k ≤ Z b a ( | f ( x ) | + | g ( x ) | ) dx = Z b a | f ( x ) | dx + Z b a | g ( x ) | dx = k f k + k g k ⇒ Hence, k f k is a norm. (ii) k x k = max 1 ≤ i ≤ n | x i | where x ∈ R n . • Proof that is satisfies Rule 1 (1.1) If x 6 = : ⇒∃ x i : | x i | > ⇒ max( | x i | ) > ⇒k x k > when x 6 = If x = : ⇒ ∀ i :...
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This note was uploaded on 09/28/2009 for the course MATH MATH427 taught by Professor Dr.sharpey during the Spring '09 term at Monmouth IL.

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sol1 - cep 2008/09 MT3802 Numerical Analysis SOLUTIONS...

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