Quotients and Products of Normed Spaces

Show that if $\mathcal{M} \leq \mathcal{X}$, then (4.1) defines a norm on $\mathcal{X}/\mathcal{M}$, where (4.1) is $\left\| x + \mathcal{M}\right\| = \inf \left\{ \left\| x+y\right\| : y \in \mathcal{M}\right\}$.

Prove that $\mathcal{X}$ is a Banach space if and only if whenever $\left\{ x_n\right\}$ is a sequence in $\mathcal{X}$ such that $\sum \left\| x_n\right\| < \infty$, then $\sum_{n=1}^{\infty } x_n$ converges in $\mathcal{X}$.

Show that if $(X, d)$ is a metric space and $\left\{ x_n\right\}$ is a Cauchy sequence such that there is a subsequence $\left\{ x_{n_k}\right\}$ that converges to $x_0$, then $x_n \to x_0$.

Find a Banach space $\mathcal{X}$ and a closed subspace $\mathcal{M}$ such that the natural map $Q : \mathcal{X} \to \mathcal{X}/\mathcal{M}$ is not a closed map. Can the natural map ever be a closed map?

Prove the converse of (4.2b) -- Theorem 4.2 states that if $\mathcal{M} \leq \mathcal{X}$ and $\left\| x + \mathcal{M}\right\|$ is defined as in (4.1), then $\left\| \cdot \right\|$ is a norm on $\mathcal{X}/\mathcal{M}$, and part (b) states that if $\mathcal{X}$ is a Banach space, then so is $\mathcal{X}/\mathcal{M}$: if $\mathcal{X}$ is a normed space, $\mathcal{M} \leq \mathcal{X}$, and both $\mathcal{M}$ and $\mathcal{X}/\mathcal{M}$ are complete, then $\mathcal{X}$ is complete. (This is an example of what is called a ``two-out-of-three'' result. If any two of $\mathcal{X}, \mathcal{M}$, and $\mathcal{X}/\mathcal{M}$ are complete, so is the third.)

Let $\mathcal{M} = \left\{ x \in l^p : x(2n) = 0 \text{ for all } n\right\}$, $1 \leq p \leq \infty$. Show that $l^p/\mathcal{M}$ is isometrically isomorphic to $l^p$.

Let $X$ be a normal locally compact space and $F$ a closed subset of $X$. If $\mathcal{M} \equiv \left\{ f \in C_0(X) : f(x) = 0 \text{ for all } x \text{ in } F\right\}$, then $C_0(X)/\mathcal{M}$ is isometrically isomorphic to $C_0(F)$.

Prove Proposition 4.4: let $\left\{ \mathcal{X}_i : i \in I\right\}$ be a collection of normed spaces and let $\mathcal{X} = \oplus_p \mathcal{X}_i$, $1 \leq p \leq \infty$, where

and $\oplus_\infty \mathcal{X}_i \equiv \left\{ x \in \prod_i \mathcal{X}_i : \left\| x\right\| \equiv \sup_i \left\| x(i)\right\| < \infty \right\}$. Then:

Formulate and prove a version of Proposition 4.4 for $\oplus_0 \mathcal{X}_n$, where, if $\left\{ \mathcal{X}_1, \mathcal{X}_2, \ldots \right\}$ is a sequence of normed spaces,

given the norm it has as a subspace of $\oplus_\infty \mathcal{X}_n$. (Proposition 4.4 states: let $\left\{ \mathcal{X}_i : i \in I\right\}$ be a collection of normed spaces and let $\mathcal{X} = \oplus_p \mathcal{X}_i$, $1 \leq p \leq \infty$. Then (a) $\mathcal{X}$ is a normed space and the projection $P_i : \mathcal{X} \to \mathcal{X}_i$ is a continuous linear map with $\left\| P_i(x)\right\| \leq \left\| x\right\|$ for each $x$ in $\mathcal{X}$; (b) $\mathcal{X}$ is a Banach space if and only if each $\mathcal{X}_i$ is a Banach space; (c) each projection $P_i$ is an open map of $\mathcal{X}$ onto $\mathcal{X}_i$.)

If $\left\{ \mathcal{X}_1, \ldots , \mathcal{X}_n\right\}$ is a finite collection of normed spaces and $1 \leq p \leq \infty$, show that the norms on $\oplus_p \mathcal{X}_k$ are all equivalent.

Here is an abstraction of Proposition 4.4. Suppose $\left\{ \mathcal{X}_i : i \in I\right\}$ is a collection of normed spaces and $Y$ is a normed space contained in $\mathbb {F}^I$. Define $\mathcal{X} \equiv \left\{ x \in \prod_i \mathcal{X}_i : \text{there is a } y \text{ in } Y \text{ with } \left\| x(i)\right\| \leq y(i) \text{ for all } i\right\}$. If $x \in \mathcal{X}$, define $\left\| x\right\| \equiv \inf \left\{ \left\| y\right\| : \left\| x(i)\right\| \leq y(i) \text{ for all } i\right\}$. Then $(\mathcal{X}, \left\| \cdot \right\| )$ is a normed space. Give necessary and sufficient conditions on $Y$ that each of the parts of (4.4) be valid for $\mathcal{X}$.

Let $\mathcal{X}$ be a normed space and $\mathcal{M} \leq \mathcal{X}$.

Let $\left\{ \mathcal{X}_i : i \in I\right\}$ be a collection of non-zero normed spaces. For $1 \leq p < \infty$, put $\mathcal{X} = \oplus_p \mathcal{X}_i$. Show that $\mathcal{X}$ is separable if and only if $I$ is countable and each $\mathcal{X}_i$ is separable. Show that $\oplus_\infty \mathcal{X}_i$ is separable if and only if $I$ is finite and each $\mathcal{X}_i$ is separable.

Show that $\oplus_0 \mathcal{X}_n$ is separable if and only if each $\mathcal{X}_n$ is separable.

Let $J \subseteq I$, and $\mathcal{X} \equiv \oplus_p \left\{ \mathcal{X}_i : i \in I\right\}$, $\mathcal{M} \equiv \left\{ x \in \mathcal{X} : x(j) = 0 \text{ for } j \text{ in } J\right\}$. Show that $\mathcal{X}/\mathcal{M}$ is isometrically isomorphic to $\oplus_p \left\{ \mathcal{X}_j : j \in J\right\}$.

Let $\mathcal{H}$ be a Hilbert space and suppose $\mathcal{M} \leq \mathcal{H}$. Show that if $Q : \mathcal{H} \to \mathcal{H}/\mathcal{M}$ is the natural map, then $Q : \mathcal{M}^\perp \to \mathcal{H}/\mathcal{M}$ is an isometric isomorphism.