Let $E$ is a infinite dimensional Banach space and $\Omega=\{x\in E: ||x||=1\}$ .
$L:\Omega\rightarrow\Omega,L(x)=-x$, how to compute the degree of $L$?
In fact ,I just know that the algebra define of degree of finite dimension and analysis define of degree of finite and infinite dimension.
Besides, why the degree of completely continue map $A:\Omega\rightarrow\Omega$ is zero ?
For many classes of Banach spaces the unit sphere is diffeomorphic to the Banach space $E$. This is a result of Bassega. So we might as well talk about mappings from $E$ to $E$.
You have to specify which degree you mean here. A classical well defined notion of degree is the Leray-Schauder degree. This is a degree for mappings of the form $$ f=I+K $$ with $I$ the identity and $K$ a compact mapping. The LS degree is defined by the degrees of finite dimensional exhaustions. To go beyond this is much more difficult. You can look at a paper of Elworthy and Tromba. They add additional structures on the Banach space to obtain a $\mathbb{Z}$ valued degree.
The problem is that most (all?) Banach spaces have a contractible general linear group. This is a result due to Kuiper. That means that we can homotope $I$ to $-I$. On a separable Hilbert space it is not difficult to write such a path explicitly. So if one insists homotopy invariance these must have the same degree. The Leray-Schauder degree is only homotopy invariant under a restricted class of homotopies (of the same type as $f$ above), which is why we can do more with such mappings.
Another degree that is useful, but kind of weak, is the Smale degree. This is well defined for mappings $E\rightarrow E$ that are proper and Fredholm. Proper Fredholm mappings have an open and dense set of regular values (this is a theorem of Smale), and the preimage of a regular value is a proper Fredholm homotopy invariant. Unfortunately it cannot distinguish $I$ from $-I$ (as they are proper Fredholm homotopic). But it can for example distinguish the mapping $f:\mathbb{R}\oplus E\rightarrow \mathbb{R} \oplus E$ defined by $f(t,x)=(t^2,x)$ from the identity.