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

of positive real numbers. We define the weighted

-space

by:

__Question 1__:

I want to show that the expression

given by:

defines a norm on

.

__Question 2:__
I now consider a special choice:

and wan't to show that

is a subspace of

.

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

to define a norm:

a)

and

b)

c)

__Showing a__
For the sequences

and

we have:

It is clear that the last inequality is satisfied only if

. Furthermore we see that

only if

. The first requirement is hereby fulfilled.

__Showing b__
For a scalar

we have that:

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 regarding

__showing of c__ and

__question 2__ . Your help is greatly appreciated.

Thank you very much.