proper invariant subspace many special cases invariant subspace problem already proven banach spaces general banach spaces complicated hilbert spaces cannot classified simple manner particular many banach spaces lack notion analogous orthonormal basis examples banach spaces math l p math spaces real number given also measure math mu math set sometimes also denoted math l p x mu math vectors equivalence classes math f math measurable functions whose absolute value 's math p math th power finite integral functions math f math
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