Let X = L²[0, 1] with 11.112 given by 1/p || f || 2 = į | f(t) |P dt ,feX. Let f and g be functions given by f(t)=t, g(t) = 1-t, Vte [0, 1]. Show that ||| f + g || ²/2 + || f- g || ²2 = 2(|| f ||²/2 + || g || ²/2 ).

Let X = L²[0, 1] with 11.112 given by 1/p || f || 2 = į | f(t) |P dt ,feX. Let f and g be functions given by f(t)=t, g(t) = 1-t, Vte [0, 1]. Show that ||| f + g || ²/2 + || f- g || ²2 = 2(|| f ||²/2 + || g || ²/2 ).

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section6.3: Trigonometric Functions Of Real Numbers

Problem 65E

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Question

Let X = L²[0, 1] with 11.112 given by
1/p
|| f || 2 =
į
| f(t) |P dt ,feX.
Let f and g be functions given by
f(t)=t, g(t) = 1-t, Vte [0, 1]. Show that
||| f + g || ²/2 + || f- g || ²2 = 2(|| f ||²/2 + || g || ²/2 ).

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