By L.A.; Sobolev, V.J. Lusternik

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**Extra resources for Elements of Functional Analysis**

**Example text**

We obtain a section of peL 9 1) which we call the "section at Infinity". Similarly, the zero section of L gives us a section of peL G 1) , which we call the zero section of P(L 9 1) . We choose a metric on L urdt circle bundle. (»( We write P vectors of length ~ 1 t o I and we let S c L be the for the part of L consisting and pro for that part of peL 9 1) c··unsisting of the section at infinity, together with all the vectors ~ uf length ltv 1T, if 0 1. We denote the projections S'" X , P ' and Since TT 00 1T 0 o ...

Uml ) 30. is the map induced by a permutation of coordinates in CmO+ml, and so is homotopic to the identity. Hence jIgl is homotopic to £1 and hence to jogo as required. Remark. It is possible to interpret vector bundles as modules in the following way. Let C(X) denote the ring of continuous complex-valued functions on X. r over X then If E is a vector bundle (E) is a C(X) - module under point-wise multiplication, i. e. , £s (x) = f(x)s (x) f € C(X) , 5 € r(E) • Moreover a homomorphism cp: E ....

Now if V has a hermitian metric h then this gives an I Homorphism V ... v* " self-dual one. and hence turns a self-conjugate bundle into We leave it as an exercise to the reader to examine In detail the symmetric+ and skew-symmetric cases and to show 'hnt, up to homotopy, the notions of self-conjugate, orthogonal, M'yrnplectic, are essentially equivalent to self-dual, real, quaternion. '11hus we ~ay pick which ever alternative is more convenient at any pH rticular stage. 3) for example to orthogonal or sympletic bundles is taot UI"e so immediate.