▶ Exercise 3.34. Suppose f1, f2: X → R are continuous functions. Their pointwise sum f₁ + f2: X → R and pointwise product fi f2: X → R are real-valued functions defined by (f1 + f2)(x) = f1(x) + f₂(x), (f1 f₂)(x) = f1(x) f₂(x). Pointwise sums and products of complex-valued functions are defined similarly. Use the characteristic property of the product topology to show that pointwise sums and products of real-valued continuous functions are continuous.
▶ Exercise 3.34. Suppose f1, f2: X → R are continuous functions. Their pointwise sum f₁ + f2: X → R and pointwise product fi f2: X → R are real-valued functions defined by (f1 + f2)(x) = f1(x) + f₂(x), (f1 f₂)(x) = f1(x) f₂(x). Pointwise sums and products of complex-valued functions are defined similarly. Use the characteristic property of the product topology to show that pointwise sums and products of real-valued continuous functions are continuous.
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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![**Exercise 3.34.** Suppose \( f_1, f_2: X \rightarrow \mathbb{R} \) are continuous functions. Their *pointwise sum* \( f_1 + f_2: X \rightarrow \mathbb{R} \) and *pointwise product* \( f_1 f_2: X \rightarrow \mathbb{R} \) are real-valued functions defined by
\[
(f_1 + f_2)(x) = f_1(x) + f_2(x),
\]
\[
(f_1 f_2)(x) = f_1(x) f_2(x).
\]
Pointwise sums and products of complex-valued functions are defined similarly. Use the characteristic property of the product topology to show that pointwise sums and products of real-valued continuous functions are continuous.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F86c8dcbb-d46d-4c91-a740-ef32ebf33ae0%2Fc2439d52-bfa9-43a8-bbc2-795e97becd2a%2F0x29arh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exercise 3.34.** Suppose \( f_1, f_2: X \rightarrow \mathbb{R} \) are continuous functions. Their *pointwise sum* \( f_1 + f_2: X \rightarrow \mathbb{R} \) and *pointwise product* \( f_1 f_2: X \rightarrow \mathbb{R} \) are real-valued functions defined by
\[
(f_1 + f_2)(x) = f_1(x) + f_2(x),
\]
\[
(f_1 f_2)(x) = f_1(x) f_2(x).
\]
Pointwise sums and products of complex-valued functions are defined similarly. Use the characteristic property of the product topology to show that pointwise sums and products of real-valued continuous functions are continuous.
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