hmwk1 - MATH 202A. HOMEWORK 1. This assignment is due Wed.,...

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Unformatted text preview: MATH 202A. HOMEWORK 1. This assignment is due Wed., Sept. 4. Some discussion is occasionally included between problem statements. In class, we defined the normed finite dimensional vectorspace ` p d as R d = { x = ( x 1 , . . . , x d ) : x j ∈ R } with the norm k x k ` p = d X j =1 | x j | p ! 1 /p and proved the H¨ older inequality d X j =1 x j y j ≤ d X j =1 | x j | p ! 1 /p d X j =1 | y j | q ! 1 /q for 1 ≤ p, q < ∞ and 1 p + 1 q = 1 We also proved the Minkowski inequality as a consequence of the H¨ older inequality: k x + y k ` p d ≤ k x k ` p d + k y k ` p d This definition can be extended to infinite sequences x = ( x 1 , x 2 , . . . ). Define k x k ` p = ∞ X j =1 | x j | p ! 1 /p We say that x ∈ ` p if this sum is finite. The space ` p so defined is an infinite- dimensional normed vector space. Indeed, the triangle inequality and H¨ older inequal- ity can be deduced from the finite dimensional versions by taking limits. We can also define ` ∞ on the set of sequences: k x k ` ∞ = sup j | x j | In the problems that follow, ` p with no subscript d means the ` p norm on the sequence space R ω , and ` p d means the ` p d norm on R d ....
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This note was uploaded on 04/06/2010 for the course MATH various taught by Professor Tao/analysis during the Spring '10 term at UCLA.

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hmwk1 - MATH 202A. HOMEWORK 1. This assignment is due Wed.,...

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