Let { f n } \{ {f_n}\} be an orthonormal system of functions on [0, 1] containing a subsystem { f n k } \{ {f_{{n_k}}}\} for which (a) f n k → 0 {f_{{n_k}}} \to 0 weakly in L 2 {L_2} , and (b) given E ⊂ [ 0 , 1 ] E \subset [0,1] , | E | > 0 |E| > 0 , Lim Inf ∫ E | f n k ( x ) | d x > 0 {\operatorname {Lim}}\;{\operatorname {Inf}}{\smallint _E}|{f_{{n_k}}}(x)|dx > 0 . There then exists a subsystem { g n } \{ {g_n}\} of { f n } \{ {f_n}\} such that for any set E as above, the linear span of { g n } \{ {g_n}\} in L 1 ( E ) {L_1}(E) is not dense. For every set E as above, there is an element of L p ( E ) {L_p}(E) , 1 > p > ∞ 1 > p > \infty , whose Walsh series expansion converges conditionally and an element of L 1 ( E ) {L_1}(E) whose Haar series expansion converges conditionally.