Given ‖ f n ‖ L 2 ≤ 2 − k , where L 2 represents the banach space of squared integrable functions, and g := ∑ k = 1 ∞ | f n | , I need to show that g ∈ L 2 ,, i.e. ∫ | g | 2 d μ &lt; ∞ .I first thought to use the Hölder inequality for f 1 = g , f 2 = g . ∫ g g d μ ≤ ( ∫ ( | f 1 | + | f 2 | + . . . ) 2 ) 1 / 2 ( ∫ ( | f 1 | + | f 2 | + . . . ) 2 ) 1 / 2 .But this is not leading to anything. It is clear that the geometric series ∑ 2 − k converges.Can somebody provide any suggestion or solution proposal ? Thanks.

Question

Given   ‖      f    n        ‖                  L        2              ≤      2          −      k        , where       L    2   represents the banach space of squared integrable functions, and   g  :=      ∑          k      =      1              ∞            |        f    n        |    , I need to show that   g  ∈      L    2    ,, i.e.   ∫      |    g            |        2    d  μ  &amp;lt;  ∞  .I first thought to use the Hölder inequality for       f    1    =  g  ,      f    2    =  g  .  ∫  g    g    d  μ  ≤  (  ∫  (      |        f    1        |    +      |        f    2        |    +  .  .  .      )    2        )          1              /            2            (  ∫  (      |        f    1        |    +      |        f    2        |    +  .  .  .      )    2        )          1              /            2        .But this is not leading to anything. It is clear that the geometric series   ∑      2          −      k       converges.Can somebody provide any suggestion or solution proposal ? Thanks.

odmeravan5c · Accepted Answer

By monotone convergence theorem,  ‖  g      ‖                  L        2              =      lim          N      →      ∞                  ‖              ∑                  k          =          1                          N                            |                    f        n                    |            ‖                      L        2              ≤      lim          N      →      ∞            ∑          k      =      1              N        ‖      f    n        ‖                  L        2              =      ∑          k      =      1              ∞        ‖  f      ‖                  L        2              &amp;lt;  ∞  .

Given <mo fence="false" stretchy="false">‖ f n </msub>

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