Question

Let $E = \{ f \mid f : [0, 1] \to \mathbb{R} \text{ integrable function} \} $ with pointwise operation. Show that $|| \cdot || : E \to \mathbb{R}$ defined by $||f|| = \int_0^1 |f(x)|dx$ is not a norm.

Name: Let E = { f | f : [0, 1] →ℝ integrable function} with pointwise operation. Show that || · || : E →ℝ defined by ||f|| = ∫0^1 |f(x)|dx is not a norm.
Uploaded: 2023-06-09T01:56:16-08:00
Duration: 4 min 17 s
Channel: Madhur L
Description: Let E = { f | f : [0, 1] →ℝ integrable function} with pointwise operation. Show that || · || : E →ℝ defined by ||f|| = ∫0^1 |f(x)|dx is not a norm.

          Let $E = \{ f \mid f : [0, 1] \to \mathbb{R} \text{ integrable function} \} $ with pointwise operation. Show that $|| \cdot || : E \to \mathbb{R}$ defined by $||f|| = \int_0^1 |f(x)|dx$ is not a norm.

$Let E = { f | f : [0, 1] →ℝ integrable function} with pointwise operation. Show that || · || : E →ℝ defined by ||f|| = ∫0^1 |f(x)|dx is not a norm.$

Added by Bego-A M.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Madhur L

06/09/2023

Step 1

||$ is not a norm, we need to show that it does not satisfy one of the properties of a norm. Show more…

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3) Let $E = \{f: [0,1] \rightarrow \mathbb{R}$ algebraic space integrable function}. with pointwise operation. Show that $||.||: E \rightarrow \mathbb{R}$ defined Consider $E$ as a vector by $||f|| = \int_{0}^{1} |f(x)|dx$, is not a norm!