Math Problems Vault

Problem 15

Let $X$ be a normed space. Prove that $X$ is a Banach space if and only if the series $\sum_{n = 1}^{\infty } a_{n}$ converges, where $(a_{n})$ is any sequence in $X$ satisfying $\sum_{n = 1}^{\infty }\left\| a_{n}\right\| < \infty$ .

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Problem 16

Let $X$ be a Banach space. Prove that the closed unit ball $\overline{B(0 ; 1)} \subset X$ is compact if and only if $X$ is finite dimensional.

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Problem 17

Let $X$ be a Banach space and $M$ a closed subspace of $X$ . Prove that the quotient space $X / M$ is also a Banach space under the quotient norm.

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Problem 18

Show that $\ell^{p}, 1 \leq p < \infty$ equipped with the norm $\left\| .\right\|_{p}$ is a Banach space.

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Problem 19

(a)(b)

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(a)

Show that $\ell^{\infty }$ equipped with the norm $\left\| .\right\|_{\infty }$ is a Banach space.

(b)

Let $c_{0}$ be the space of sequences converging to 0. Show that $c_{0}$ is a closed subspace of $\ell^{\infty }$ .

Problem 20

(a)(b)

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(a)

Let $c_{00}$ be the space of sequences such that if $x = \left(x_{n}\right)_{n \in \mathbb {N}} \in c_{00}$ then $x_{n} = 0$ for all $n \geq n_{0}$ , where $n_{0}$ is some integer number. Show that $c_{00}$ with the norm $\left\| .\right\|_{\infty }$ is NOT a Banach space.

(b)

What is the closure of $c_{00}$ in $\ell^{\infty }$ ?

Problem 21

(a)(b)

Prove that:

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(a)

If $1 \leq p < q < \infty$ , then $\ell^{p} \subset \ell^{q}$ and $\left\| x\right\|_{q} \leq \left\| x\right\|_{p}$ .

(b)

If $x \in \bigcup_{1 \leq p < \infty } \ell^{p}$ then $\left\| x\right\|_{p} \rightarrow \left\| x\right\|_{\infty }$ as $p \rightarrow \infty$ .

Problem 22

(a)(b)

Prove that:

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(a)

If $1 \leq p < \infty$ then $\ell^{p}$ is separable.

(b)

$\ell^{\infty }$ is not separable.

Problem 23

(a)(b)

Let $C[0,1]$ be the space of all continuous functions on $[0,1]$ .

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(a)

Prove that if $C[0,1]$ is equipped with the uniform norm

\left\| f\right\| = \max _{x \in [0,1]}\left|f(x)\right|, \quad f \in C[0,1]

then $C[0,1]$ is a Banach space.

(b)

Give an example to show that $C[0,1]$ equipped with the $L^{1}$ -norm

\left\| f\right\| _{1} = \int _{0}^{1}\left|f(x)\right| d x, \quad f \in C[0,1]

is not a Banach space.