# Thread: How to show that something is a subspace of something else?

1. ## How to show that something is a subspace of something else?

Hi

I know that this is a basic question but it is one that I cannot seem to be able to get my head around. If I have a set

and another set where the thing condition that there can only be finitely many x_k different from zero is replaced by the condition that the supremum of x_k must be finite.

How can I show that the second set is a subspace of the first set?

I know about the addition and multiplication with scalar, but I just cannot seem to be able to understand it so I was hoping for a thorough example.

Thanks a lot

2. Originally Posted by Ase
Hi

I know that this is a basic question but it is one that I cannot seem to be able to get my head around. If I have a set

and another set where the thing condition that there can only be finitely many x_k different from zero is replaced by the condition that the supremum of x_k must be finite.

How can I show that the second set is a subspace of the first set?

I know about the addition and multiplication with scalar, but I just cannot seem to be able to understand it so I was hoping for a thorough example.

Thanks a lot
The first thing you need to show is that the first set is a subset of the second. That is true because, if there are only a finite number of non-zero $x_k$s, there is a largest one which is the finite supremum.

The second thing you must show is that the set is closed under addition and scalar multiplication. I am assuming "coordinate-wise" addition and scalar multiplication here: $\{x_k\}+ \{y_k\}= \{x_k+ y_k\}$ and $r\{x_k\}= \{rx_k\}$. Scalar multiplication is easy: all but a finite number of the $x_k$ are 0 and any scalar times 0 is 0. So all but a finite number of $rx_k$ are 0.

For addition, suppose $\{x_k\}$ has n non-zero entries and $\{y_k\}$ has m non-zero entries. Can you see that $\{x_k+ y_k\}$ has, at most, m+n non-zero entries?

3. Again you are a lifesaver HallsofIvy. I have been trying to get people to explain it to me in ways in understand so many times. The only difference here is that you have made it understandable.

Thanks a million