- #1
maze
- 662
- 4
In finite dimensions, a matrix can be decomposed into the sum of rank-1 matrices. This got me thinking - in what situations can a bounded linear operator mapping between infinite dimensional spaces be written as an (infinite) sum of rank-1 operators?
eg, let A be a bounded linear operator from banach spaces X to Y, then perhaps we might try
[tex]A = \sum_{i=1}^\infty y_i \phi_i[/itex]
for some functionals [itex]\phi_i[/itex] in X', and elements y_i in Y.
Is there anything to this idea?
eg, let A be a bounded linear operator from banach spaces X to Y, then perhaps we might try
[tex]A = \sum_{i=1}^\infty y_i \phi_i[/itex]
for some functionals [itex]\phi_i[/itex] in X', and elements y_i in Y.
Is there anything to this idea?