A unitary representation is completely reducible, in the sense that for any closed invariant subspace, the orthogonal complement is again a closed invariant subspace.
22.
Note that this is the typical formulation of an eigenvalue problem, which means that any eigenvector of A forms a uni-dimensional invariant subspace in T
23.
He says, I think, if a commuting family leaves any subspace invariant, then that commuting family has a common eigenvector in that invariant subspace.
24.
The topic of " invariant subspaces " is a coherent topic in operator theory ( functional analysis ), which is described in many monographs and textbooks.
25.
Moreover, every vector v \ in V is either an eigenvector of f, or is in an invariant subspace with respect to f of dimension two.
26.
When reduced modulo 3 this has 1-dimensional invariant subspaces and quotient spaces, giving an irreducible representation of dimension 781 over the field with 3 elements.
27.
For certain linear operators there is no " non-trivial " invariant subspace; consider for instance a rotation of a two-dimensional real vector space.
28.
Comparing with the previous example, one can see that the invariant subspaces of a linear transformation are dependent upon the underlying scalar field of " V ".
29.
As the above examples indicate, the invariant subspaces of a given linear transformation " T " shed light on the structure of " T ".
30.
For Banach spaces, the first example of an operator without an invariant subspace was constructed by Enflo . ( For Hilbert spaces, the invariant subspace problem remains open .)
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