A What do the notations in functional analysis mean for a given function?

[member 428835]

Hi PF!

Can someone help me understand the notation here (I've looked everywhere but can't find it): given a function $f:G\to \mathbb R$ I'd like to know what $C(G),C(\bar G),L_2(G),W_2^1(G),\dot W_1^2(G)$. I think $C(G)$ implies $f$ is continuous on $G$ and that $C(\bar G)$ implies $f$ is continuous on $\bar G$. $L_2(G)$ implies $F$ is square-integrable on $G$, but I'm not sure of $W_2^1(G),\dot W_2^1(G)$.

Any help would be awesome!

 

[member 587159]

Hi PF!

Can someone help me understand the notation here (I've looked everywhere but can't find it): given a function $f:G\to \mathbb R$ I'd like to know what $C(G),C(\bar G),L_2(G),W_2^1(G),\dot W_1^2(G)$. I think $C(G)$ implies $f$ is continuous on $G$ and that $C(\bar G)$ implies $f$ is continuous on $\bar G$. $L_2(G)$ implies $F$ is square-integrable on $G$, but I'm not sure of $W_2^1(G),\dot W_2^1(G)$.

Any help would be awesome!

I think you are right about the C and the L notations, although it is hard to say without context. I have never seen the W notation though...

 

[fresh_42]

I have a book in which $W_{2,k}(a,b)$ is defined as all (complex valued) functions on $[a,b]$ which are $k-1$ fold continuously differentiable, where the $k-1$ derivative is absolutely continuous and the $k-$th derivative is in $L_2(a,b)$ with
$$
\langle f,g\rangle_k = \sum_{j=0}^k \int_a^b f^{(j)}(x)^*g^{(j)}(x)\,dx
$$
$W_{2,s}(\mathbb{R}^m)$ is defined as the Sobolev space of order $s$ in the book. Tao also uses $W$ for Sobolev spaces.

But I don't think one can just use those terms without definition.

 

[member 428835]

Thank you both! I found a definition through generalized derivatives.