Hello,

I have been trying to solve two questions concerning norm-defining and subspace. This I am trying to do with respect to Banach-spaces. Allow me to introduce the problem (it's a bit long so bear with me):

I am considering a sequence $\displaystyle \{w_{k}\}_{k=1}^{\infty} $ of positive real numbers. We define the weighted $\displaystyle \ell^{1}$-space $\displaystyle \ell_{w}^{1}(\mathbb{N}) $ by:

$\displaystyle \ell_{w}^{1}(\mathbb{N}) = \left\lbrace \{x_{k} \}_{k=1}^{\infty} \mid x_{k} \in \mathbb{C}, \sum_{k=1}^{\infty} |x_{k}|w_{k} < \infty \right\rbrace $

:Question 1

I want to show that the expression $\displaystyle \Vert \cdot \Vert $ given by:

$\displaystyle \Vert \{x_{k} \}_{k=1}^{\infty} \Vert = \sum_{k=1}^{\infty} |x_{k}|w_{k} $

defines a norm on $\displaystyle \ell_{w}^{1}(\mathbb{N}) $.

Question 2:

I now consider a special choice:

$\displaystyle w_{k}=2^{k}$

and wan't to show that $\displaystyle \ell_{w}^{1}(\mathbb{N})$ is a subspace of $\displaystyle \ell^{1}(\mathbb{N})$.

The following is what I have done so far:

:Solution to question 1

The following three condition must be fulfilled in order for a function $\displaystyle \Vert \cdot \Vert $ to define a norm:

a) $\displaystyle \Vert \textbf{v} \Vert \geq 0, \forall \textbf{v} \in V$ and $\displaystyle \Vert \textbf{v} \Vert=0 \Leftrightarrow \textbf{v}=\textbf{0}$

b) $\displaystyle \Vert \alpha \textbf{v}\Vert = |\alpha|\Vert \textbf{v}\Vert, \forall \textbf{v} \in V, \alpha \in \mathbb{C} $

c) $\displaystyle \Vert \textbf{v}+\textbf{w}\Vert \leq \Vert \textbf{v} \Vert + \Vert \textbf{w} \Vert, \forall \textbf{v},\textbf{w} \in V$

Showing a

For the sequences $\displaystyle \{x_{k}\}_{k=1}^{\infty} $ and $\displaystyle \{w_{k}\}_{k=1}^{\infty} $ we have:

$\displaystyle \Vert \{x_{k} \}_{k=1}^{\infty} \Vert \geq 0 \Leftrightarrow

\sum_{k=1}^{\infty} |x_{k}|w_{k} \geq 0 \Leftrightarrow

|x_{1}|w_{1} + |x_{2}|w_{2} + \ldots + |x_{k}|w_{k} + \cdots \geq 0$

It is clear that the last inequality is satisfied only if $\displaystyle \{x_{k}\}_{k=1}^{\infty},\{w_{k}\}_{k=1}^{\infty} \neq \{0\}_{k=1}^{\infty}$. Furthermore we see that $\displaystyle \Vert \{x_{k} \}_{k=1}^{\infty} \Vert=0$ only if $\displaystyle \{x_{k} \}_{k=1}^{\infty}=\{w_{k} \}_{k=1}^{\infty}=\{ 0\}_{k=1}^{\infty}$. The first requirement is hereby fulfilled.

Showing b

For a scalar $\displaystyle \alpha \in \mathbb{C}$ we have that:

$\displaystyle |\alpha| \cdot \Vert \{x_{k} \}_{k=1}^{\infty} \Vert = $

$\displaystyle |\alpha|\cdot \sum_{k=1}^{\infty} |x_{k}|w_{k} = $

$\displaystyle |\alpha|\cdot ( |x_{1}|w_{1} + |x_{2}|w_{2} + \ldots + |x_{k}|w_{k} +\cdots ) =$

$\displaystyle |\alpha||x_{1}|w_{1} + |\alpha||x_{2}|w_{2} + \ldots + |\alpha||x_{k}|w_{k} + \cdots =$

$\displaystyle \sum_{k=1}^{\infty} |\alpha||x_{k}|w_{k} =$

$\displaystyle \Vert \alpha \{x_{k} \}_{k=1}^{\infty} \Vert $

Requirement 2 is hereby met.

Showing c

Having trouble. Assistance needed.

Solution to question 2

Having trouble. Help needed.

Sorry for the long post but I hope that someone can be of assistance regardingshowing of cand. Your help is greatly appreciated.question 2

Thank you very much.