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 f k = Z b a | f ( x ) | dx = Z b a | || f ( x ) | dx = | | Z b a | f ( x ) | dx = | |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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sol1 - cep 2008/09 MT3802 - Numerical Analysis SOLUTIONS -...

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