Proof of a normed space

[cabin5]

Homework Statement

The norm is defined by $$\left\|x\right\|_{p}=\left[\sum^{\infty}_{k=1}\left|\alpha^{i}\right|^{p}\right]^{1/p}$$
where $$x=(\alpha^{1},\alpha^{2},...,\alpha^{n})$$

Prove that this is a norm on $$V_{\infty}(F)$$

Homework Equations

All conditions satisfied for a normed space.

The Attempt at a Solution

Well, I proved the first condition which is

i)
$$\left\|x\right\|_{p}=\left\{\left|\alpha^{1}\right|^{p}+\left|\alpha^{2}\right|^{p}+...+\left|\alpha^{n}\right|^{p}+...\right\}^{1/p}$$

This must be positive definitive, therefore $$\left\|x\right\|_{p}>0$$

On the second condition I don't know whether taking the product of this norm with a $$\beta\in F$$ since the sum is infinite. I got stuck at this point.

And also I presume, for the third condition, Minkowski inequality cannot be used anymore to prove it.

 

[mighty2000]

Any help would be greatly appreciated!

it is important to understand the fundamental properties and definitions of mathematical concepts in order to apply them accurately and effectively in research. In this case, we are looking at the norm defined by \left\|x\right\|_{p}=\left[\sum^{\infty}_{k=1}\left|\alpha^{i}\right|^{p}\right]^{1/p}.

To prove that this is a norm on V_{\infty}(F), we need to show that it satisfies the three conditions for a normed space: positivity, absolute homogeneity, and the triangle inequality.

For the first condition, we can see that \left\|x\right\|_{p} is always positive since it is a sum of non-negative terms raised to the power of p and then taking the pth root. This means that \left\|x\right\|_{p}>0 for all x\in V_{\infty}(F).

Next, for the second condition of absolute homogeneity, we need to show that \left\|\beta x\right\|_{p}=|\beta|\left\|x\right\|_{p} for all x\in V_{\infty}(F) and \beta\in F. Since the sum is infinite, we cannot simply take the product of the norm with \beta. Instead, we can use the fact that raising a number to the power of p and then taking the pth root is the same as taking the absolute value. Therefore, we have:

\left\|\beta x\right\|_{p}=\left[\sum^{\infty}_{k=1}\left|\beta\alpha^{i}\right|^{p}\right]^{1/p}=\left[\sum^{\infty}_{k=1}|\beta|^{p}\left|\alpha^{i}\right|^{p}\right]^{1/p}=|\beta|\left[\sum^{\infty}_{k=1}\left|\alpha^{i}\right|^{p}\right]^{1/p}=|\beta|\left\|x\right\|_{p}

This shows that the norm is absolutely homogeneous.

Finally, for the third condition of the triangle inequality, we need to show that \left\|x+y\right\|_{p}\leq\left