Given f k ∈ L p ( Ω ), I have to prove that ‖ ∑ n = 1 ∞ f n ‖ L p ( Ω ) ⩽ ∑ n = 1 ∞ ‖ f n ‖ L p ( Ω ) , where ‖ f ‖ ≐ ( ∫ Ω | f | p ) 1 / p denotes the p-norm, with 1 ⩽ p &lt; ∞. Since I've already proved the original Minkowski inequality, a simple induction estabilishes the finite case, i.e., ‖ ∑ n = 1 m f n ‖ L p ( Ω ) ⩽ ∑ n = 1 m ‖ f n ‖ L p ( Ω ) ( ∗ ), N ∋ m &lt; ∞.So, what I've thought is, since L p ( Ω ) is a Banach space, to suppose that ∑ n = 1 ∞ ‖ f n ‖ L p ( Ω ) &lt; ∞, which would give us that ∑ n = 1 ∞ f n converges in L p ( Ω ), by completeness. Hence, the desired inequality follow by making m ⟶ ∞ in ( ∗ ), but I'm not completely sure about this.Any help will be appreciated.

Question

Given       f    k    ∈      L    p    (  Ω  ), I have to prove that             ‖              ∑                  n                    =                    1                ∞                    f        n            ‖                      L        p            (      Ω      )        ⩽      ∑          n            =            1        ∞              ‖              f        n            ‖                      L        p            (      Ω      )      , where       ‖    f    ‖    ≐            (                        ∫          Ω                                      |            f            |                    p                    )              1              /            p       denotes the p-norm, with   1  ⩽  p  &amp;lt;  ∞. Since I&#039;ve already proved the original Minkowski inequality, a simple induction estabilishes the finite case, i.e.,             ‖              ∑                  n                    =                    1                m                    f        n            ‖                      L        p            (      Ω      )        ⩽      ∑          n            =            1        m              ‖              f        n            ‖                      L        p            (      Ω      )        (  ∗  ),       N    ∋  m  &amp;lt;  ∞.So, what I&#039;ve thought is, since       L    p    (  Ω  ) is a Banach space, to suppose that       ∑          n            =            1        ∞              ‖              f        n            ‖                      L        p            (      Ω      )        &amp;lt;  ∞, which would give us that       ∑          n            =            1        ∞        f    n   converges in       L    p    (  Ω  ), by completeness. Hence, the desired inequality follow by making   m  ⟶  ∞ in   (  ∗  ), but I&#039;m not completely sure about this.Any help will be appreciated.

Zayden Wiley · Accepted Answer

There is really nothing difficult. What you know is that      g    n        ∑          i      =      1        n        f    n  converges to some element   g in       L    p  . In particular,   ‖      g    n    ‖  →  ‖  g  ‖. Since  ‖      g    n    ‖  =      ‖          ∑              i        =        1            n              f      n        ‖    ≤      ∑          i      =      1        n    ‖      f    n    ‖  ≤      ∑          i      =      1        ∞    ‖      f    n    ‖The sequence of real numbers   (  ‖      g    n    ‖      )          n      =      1        ∞   is uniformly bounded by       ∑          i      =      1        ∞    ‖      f    n    ‖, this implies  ‖  g  ‖  ≤      ∑          i      =      1        ∞    ‖      f    n    ‖  .Note that   g  =      ∑          i      =      1        n        f    n  . Thus you are done.

Given f k </msub> ∈ L p </msup> ( <mi m

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