Definition of $\ell^p$ space and some confusions with norm

Question

$\ell^{p}$ spaces [edit]

See also: $L^{p}$ space and $L$ -infinity

For $0<p<\infty, \ \ell^{p}$ is the subspace of $\mathbb{K}^{\mathbb{N}}$ consisting of all sequences $x=\left(x_{n}\right)$ satisfying $$ \sum_{n}\left|x_{n}\right|^{p}<\infty $$ If $p \geq 1,$ then the real-valued operation $\|\cdot\|_{p}$ defined by $$ \|x\|_{p}=\left(\sum_{n}\left|x_{n}\right|^{p}\right)^{1 / p} $$ defines a norm on $\ell^{p} .$ In fact, $\ell^{p}$ is a complete metric space with respect to this norm, and therefore is a Banach space. If $0<p<1,$ then $\ell^{p}$ does not carry a norm, but rather a metric defined by $$ d(x, y)=\sum_{n}\left|x_{n}-y_{n}\right|^{p} $$

Above is transcribed from wikipedia(screenshot here), from my understanding, $\ell^p$ is the subspace of sequence space which all terms summation to the power of $p$ is finite. In this case, $p$-norm is make sense, which analogous from Euclidean distance. However,

Question 1: what is the point of calculating such sequence with Euclidean norm, it just not make sense in higher dimension.

Quesiton 2: I don't understand why does when $0<p<1$, it's not carry a norm, stated in last line from capture.

Please enlighten me, any help would be appreciated.

Answer 1

If $0<p<1$ then the triangular inequality does not hold for $\|\cdot\|_p$. For instance, take $\ell^{1/2}(\mathbb R)$ and consider the elements

$$x=(1/4, 0, 0, \ldots), y=(0,1/4 , 0, 0, \ldots)\in \ell^{1/2}(\mathbb R).$$

Then

$$\|x+y\|_{1/2}=\left(\left|\frac{1}{4}\right|^{1/2}+\left|\frac{1}{4}\right|^{1/2}\right)^{2}=\left(\frac{1}{2}+\frac{1}{2} \right)^2=1^2=1. $$

On the other hand:

$$\|x\|_{1/2}+\|y\|_{1/2}=\left(\left|\frac{1}{4}\right|^{1/2}\right)^2+\left(\left|\frac{1}{4}\right|^{1/2}\right)^2=\frac{1}{4}+\frac{1}{4}=\frac{1}{2}.$$ Therefore:

$$\|x\|_{1/2}+\|y\|_{1/2}<\|x+y\|_{1/2}.$$

Answer 2

Question 2: compare $d(x,y)$ with $d(x,0)=d(y,0)$, where $x = (1,0,0,\dots)$ and $y_n = (0,1,0,\dots)$. Note that this is a finite dimensional counterexample i.e. basically the same problem occurs for $\mathbb R^2$ equipped with $d(x,y)^p = |x_1-y_1|^p + |x_2-y_2|^p$.

Re: question 1, Firstly, no one ever said that a definition needs a point. I can of course make the definition $\texttt{LJNG}:=42$. Whats the point? Who knows. It does make sense, however. And here there is a point, the $\ell^p$ spaces are basic examples of Banach spaces which are very important and heavily studied. Having examples is good because the theory is hard and intuition from finite dimensions do not carry over.