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Exercise 1.2.5 Show that for 1 ≤ p ≤ ∞, ||x|| is a norm in the space Rd. The main task is to veryy the triangle inequality, which can be done by first proving the Hölder inequality, x·y| ≤ ||x||p||y||p', x, y e Rd. Here p' is the conjugate of p defined through the relation 1/p' + 1/p = 1; by convention, p' = 1 if p = ∞, p' = ∞ if p = 1.

Exercise 1.2.5 Show that for 1 ≤ p ≤ ∞, ||x|| is a norm in the space Rd. The main task is to veryy the triangle inequality, which can be done by first proving the Hölder inequality, x·y| ≤ ||x||p||y||p', x, y e Rd. Here p' is the conjugate of p defined through the relation 1/p' + 1/p = 1; by convention, p' = 1 if p = ∞, p' = ∞ if p = 1.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Exercise 1.2.5 Show that for 1 ≤p ≤ ∞, ||x||p is a norm in the space Rd. The main task is to veryy the triangle inequality, which can be done by first proving the Hölder inequality, |x-y| ≤ ||x||p|ly||p', x,y e Rd. Here p' is the conjugate of p defined through the relation 1/p' + 1/p= 1; by convention, p′ = 1 if p = ∞, p′ = ∞ if p = 1.
Transcribed Image Text:Exercise 1.2.5 Show that for 1 ≤p ≤ ∞, ||x||p is a norm in the space Rd. The main task is to veryy the triangle inequality, which can be done by first proving the Hölder inequality, |x-y| ≤ ||x||p|ly||p', x,y e Rd. Here p' is the conjugate of p defined through the relation 1/p' + 1/p= 1; by convention, p′ = 1 if p = ∞, p′ = ∞ if p = 1.
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