Let (Xn) be a sequence of normed spaces and let X0={x ∈∏n ∈ℕ Xn:xn 0=0 人 } be their c0-sum (wit

Step 1: Understand the definition of the space u005C( X_0 u005C). The space u005C( X_0 u005C) is defined as the

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Let $\left(X_n\right)$ be a sequence of normed spaces and let $$ X_0=\left\{x \in \prod_{n \in \mathbb{N}} X_n:\left\|x_n\right\| 0=0 \text { 人 }\right\} $$ be their $c_0$-sum (with the norm $\|x\|=\sup \left\{\left\|x_n\right\|: n \in \mathbb{N}\right\}$ induced from the $l_{\infty}$-sum). Prove that $X_0$ is separable if and only if so is each of the spaces $X_n$. 5.19. Prove that the space $C^{(p)}[0,1]$ presents the sum of a finite-dimensional subspace and a space isomorphic to $C[0,1]$.

    Let $\left(X_n\right)$ be a sequence of normed spaces and let
$$
X_0=\left\{x \in \prod_{n \in \mathbb{N}} X_n:\left\|x_n\right\| 0=0 \text { 人 }\right\}
$$
be their $c_0$-sum (with the norm $\|x\|=\sup \left\{\left\|x_n\right\|: n \in \mathbb{N}\right\}$ induced from the $l_{\infty}$-sum). Prove that $X_0$ is separable if and only if so is each of the spaces $X_n$.
5.19. Prove that the space $C^{(p)}[0,1]$ presents the sum of a finite-dimensional subspace and a space isomorphic to $C[0,1]$.

Fundamentals of Functional Analysis

S. S. Kutateladze 1st Edition

Chapter 5, Problem 18

Instant Answer

Step 1

The space $ X_0 $ is defined as the set of all sequences $ x = (x_n) $ where $ x_n \in X_n $ and $ \|x_n\| \to 0 $ as $ n \to \infty $. The norm on $ X_0 $ is given by $ \|x\| = \sup \{\|x_n\| : n \in \mathbb{N}\} $. Show more…

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Let $\left(X_n\right)$ be a sequence of normed spaces and let $$ X_0=\left\{x \in \prod_{n \in \mathbb{N}} X_n:\left\|x_n\right\| 0=0 \text { 人 }\right\} $$ be their $c_0$-sum (with the norm $\|x\|=\sup \left\{\left\|x_n\right\|: n \in \mathbb{N}\right\}$ induced from the $l_{\infty}$-sum). Prove that $X_0$ is separable if and only if so is each of the spaces $X_n$. 5.19. Prove that the space $C^{(p)}[0,1]$ presents the sum of a finite-dimensional subspace and a space isomorphic to $C[0,1]$.

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