Understanding the P-norm. I understand that a sequence $x=(x_1,x_2,x_3,\ldots)=(x_k)_{k \in K} $ where $x_k \in \mathbb{ K}$ for all $k \in \mathbb{N}$    belongs to the sequence space $l_{p} (\mathbb{N})$
if $$||x||_{p}=\bigg(\sum^\infty_{k=1} |x_{k}|^p \bigg)^\frac{1}{p} < \infty$$ 
I also know that when $P=1$, we get the "taxi cab" norm and when $P=2$, we get the Euclidean norm. 
What do we get when $P>2$?
In my notes it also says a sequence $x=(x_1,x_2,x_3,\ldots)=(x_k)_{k \in K} $ where $x_k \in \mathbb{ K}$ for all $k \in \mathbb{N}$    belongs to the sequence space $l_{p} (\mathbb{N})$ if $$||x||_{\infty}=sup_{k \in \mathbb{N}} |x_k|$$
Why are there two expressions defining when a sequence belongs to a sequence space?  
Intuitively what do we mean when we say that a sequence belongs to a sequence space?  
Lastly is there a difference between $l^p$ and $L^p$? 
 A: $l^p $ is a special case of $L^p$.
Let $f:X\rightarrow \mathbb{C}$ and define $||f||_p= \ (\int_X |f|^p d \mu \ )^{ \frac{1}{p}}$
We know that, $L^p( \mu)$ consists of all $f$ such that $||f||_p < \infty$.
The special case is- 
$\mu $ is the counting measure and the space X is $\mathbb{N}$. Hence $f:X \rightarrow \mathbb{C}$ changes to $f:\mathbb{N} \rightarrow \mathbb{C}$ which is nothing but a complex sequence.
Also because of the counting measure $\int(.)$ changes to $\sum(.)$
A: When $1<p<2$, or when $p>2$, you just get a different norm. Different in that it actually defines a different topology, i.e. you can come up with sequences of sequences that converge in one norm and not in the other one. 
The assertion in your notes is wrong. The equality $\|x\|_\infty=\sup|x_k|$ defines a further norm the infinity norm. Not every sequence with $\|x\|_\infty<\infty$ is in $\ell^p(\mathbb N)$: for instance consider the constant sequence $x=(1,1,\ldots)$. Then $\|x\|_\infty=1$, while $x\not\in\ell^p(\mathbb N)$ for any $p\in[1,\infty)$. 
Finally, $\ell^p(\mathbb N)$ is $L^p(\mathbb N,\Omega,\mu)$, where $\Omega$ is the power set of $\mathbb N$ and $\mu$ is the counting measure. 
