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Let E be a Euclidean space with inner ratic form. For all u, v E E, we have the C y(u, v)² ≤ $(u)(v), if u and v are linearly dependent.

Let E be a Euclidean space with inner ratic form. For all u, v E E, we have the C y(u, v)² ≤ $(u)(v), if u and v are linearly dependent.

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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Let E be a Euclidean space with inner product y, and let & be the corresponding quadratic form. For all u, v E E, we have the Cauchy-Schwarz inequality y(u, v)² ≤ Ⓡ(u)Ⓡ(v), the equality holding iff u and u are linearly dependent. We also have the Minkowski inequality √®(u + v) ≤ √0(u) + √(v), the equality holding iff u and u are linearly dependent, where in addition if u 0 and v/ 0, then u = Au for some >> 0.
Transcribed Image Text:Let E be a Euclidean space with inner product y, and let & be the corresponding quadratic form. For all u, v E E, we have the Cauchy-Schwarz inequality y(u, v)² ≤ Ⓡ(u)Ⓡ(v), the equality holding iff u and u are linearly dependent. We also have the Minkowski inequality √®(u + v) ≤ √0(u) + √(v), the equality holding iff u and u are linearly dependent, where in addition if u 0 and v/ 0, then u = Au for some >> 0.
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