Let Y R([a, b]) be the set (actually a vector space) of bounded, real-valued, Riemann integrable functions on the closed and bounded interval [a, b], and let X = C([a, b]), be the subset (actually a vector subspace) of continuous, real-valued functions on [a, b]. For f, g € Y, define d(f,g) sup f(x) = g(x)\. RElab = Show that the function dxxx X x X has the following proper- ties: for all f, g, h EX, i. d(f,g) ≥ 0 and d(f, g) = 0 if and only if f = g (i.e. if and only if f(x) = g(x) for all x € [a, b].); ii. d(f, g) = d(g, f); iii. d(f,g) ≤d(f, h) + d(h, g).
Let Y R([a, b]) be the set (actually a vector space) of bounded, real-valued, Riemann integrable functions on the closed and bounded interval [a, b], and let X = C([a, b]), be the subset (actually a vector subspace) of continuous, real-valued functions on [a, b]. For f, g € Y, define d(f,g) sup f(x) = g(x)\. RElab = Show that the function dxxx X x X has the following proper- ties: for all f, g, h EX, i. d(f,g) ≥ 0 and d(f, g) = 0 if and only if f = g (i.e. if and only if f(x) = g(x) for all x € [a, b].); ii. d(f, g) = d(g, f); iii. d(f,g) ≤d(f, h) + d(h, g).
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.4: Linear Transformations
Problem 34EQ
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![Let Y= R([a, b]) be the set (actually a vector space) of bounded,
real-valued, Riemann integrable functions on the closed and bounded
interval [a, b], and let X = C([a, b]), be the subset (actually a vector
subspace) of continuous, real-valued functions on [a, b]. For f, g € Y,
define
d(f,g) sup f(x) = g(x)|.
x= [a,b]
Show that the function
ties: for all f, g, h € X,
dxxx: xXx X has the following proper-
i. d(f,g) ≥ 0 and d(f, g) = 0 if and only if f = g (i.e. if and
only if f(x) = g(x) for all x = [a, b].);
ii.
d(f,g) = d(g, f);
iii. d(f,g) ≤d(f, h) + d(h,g).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F37e8ed93-7bef-4409-89ed-52264f64a27e%2F91a973af-a27b-4e65-ae36-bf8eb68a54bc%2F8qcczyl_processed.png&w=3840&q=75)
Transcribed Image Text:Let Y= R([a, b]) be the set (actually a vector space) of bounded,
real-valued, Riemann integrable functions on the closed and bounded
interval [a, b], and let X = C([a, b]), be the subset (actually a vector
subspace) of continuous, real-valued functions on [a, b]. For f, g € Y,
define
d(f,g) sup f(x) = g(x)|.
x= [a,b]
Show that the function
ties: for all f, g, h € X,
dxxx: xXx X has the following proper-
i. d(f,g) ≥ 0 and d(f, g) = 0 if and only if f = g (i.e. if and
only if f(x) = g(x) for all x = [a, b].);
ii.
d(f,g) = d(g, f);
iii. d(f,g) ≤d(f, h) + d(h,g).
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