I was reading the definition of Sobolev space form book Evans.
From that, I understand the following,
$W^{k,p}(U)$ is space of locally integrable function whose all derivative up to order k are in $L^p$.
So $W^{k,p}(U)\subset L_{loc}^1(U)\cap L^p(U)$.
I thought locally summable condition is considered for defining weak derivative.
Is my understanding of Sobolev space is correct? Please tell me. As I am learning independently with the only help of Mathstack. Please Help me.
Any Help will be appreciated.
The answer is yes. It is enough to notice that if a function $f$ belongs to $L^p(U)$, then it immediately follows that $$ f\in L^p_{loc}(U). $$ Since $p\geq 1$ (you didn't write this but I assumed you are working with $p\geq1$), and by using the standard inclusions of $L^p$ spaces on compact domains we know that $$ L^p_{loc}(U)\subset L^1_{loc}(U). $$ Hence, it follows that, for any $k\in\mathbb{N}$ and any $p\geq 1$ we have $$ W^{k,p}(U)\subset L^1_{loc}(U)\cap L^p(U). $$