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Let (X, || ||x) and (Y, ||- ||y) be the normed spaces. Let XOY := {(x, y): xe X;y ≤ Y} denote the direct sum of X and Y. For each element (x, y) E XY, put ||(x, y)||₁ := ||||x+||yl|y. (a) Show that || ||₁ is a norm function on XY. . (b) Show that if X and Y both are Banach spaces then the space X Y under the norm || ||1 is also a Banach space.

Let (X, || ||x) and (Y, ||- ||y) be the normed spaces. Let XOY := {(x, y): xe X;y ≤ Y} denote the direct sum of X and Y. For each element (x, y) E XY, put ||(x, y)||₁ := ||||x+||yl|y. (a) Show that || ||₁ is a norm function on XY. . (b) Show that if X and Y both are Banach spaces then the space X Y under the norm || ||1 is also a Banach space.

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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1. Let (X, |-|x) and (Y, |-|y) be the normed spaces. Let XOY:= {(x, y): xe X; y € Y} denote the direct sum of X and Y. For each element (x, y) = X Y, put ||(x, y) ||1= ||||x + ||yl|y. (a) Show that || ||₁ is a norm function on XY. . (b) Show that if X and Y both are Banach spaces then the space X Y under the norm | ||₁ is also a Banach space.
Transcribed Image Text:1. Let (X, |-|x) and (Y, |-|y) be the normed spaces. Let XOY:= {(x, y): xe X; y € Y} denote the direct sum of X and Y. For each element (x, y) = X Y, put ||(x, y) ||1= ||||x + ||yl|y. (a) Show that || ||₁ is a norm function on XY. . (b) Show that if X and Y both are Banach spaces then the space X Y under the norm | ||₁ is also a Banach space.
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