$L_d = { phi in L_{[-a,a]}^2 vert phi(t) = -phi(-t) }$, prove that is a closed subspace of $L^2_{[-a,a]}$












2














$L_d = { phi in L_{[-a,a]}^2$ $vert$ $phi(t) = -phi(-t)$ $forall t in [-a,a] }$, prove that is a closed subspace of $L^2_{[-a,a]}$.



Now, the subspace part is always the same and I managed.



The part on closeness is where I stumble. This is what I did:



I want to prove it using closeness by sequences, so:



Let ${phi_n} subset L_d$ be a sequence, such that: $phi_n to phi$ in $L^2$. I want to prove that $phi in L_d$.



I know that $phi_n to phi$ in $L^2$ implies that there exists an extracted subsequence $phi_{n_k}$ such that $phi_{n_k} to phi'$ almost everywhere. I also know that $phi' = phi$ almost everywhere for uniqueness of the limit. But still, I do not know how to say that $phi' = phi in L^d$. I feel like I am just missing this last step but I am not sure on how to proceed.










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    2














    $L_d = { phi in L_{[-a,a]}^2$ $vert$ $phi(t) = -phi(-t)$ $forall t in [-a,a] }$, prove that is a closed subspace of $L^2_{[-a,a]}$.



    Now, the subspace part is always the same and I managed.



    The part on closeness is where I stumble. This is what I did:



    I want to prove it using closeness by sequences, so:



    Let ${phi_n} subset L_d$ be a sequence, such that: $phi_n to phi$ in $L^2$. I want to prove that $phi in L_d$.



    I know that $phi_n to phi$ in $L^2$ implies that there exists an extracted subsequence $phi_{n_k}$ such that $phi_{n_k} to phi'$ almost everywhere. I also know that $phi' = phi$ almost everywhere for uniqueness of the limit. But still, I do not know how to say that $phi' = phi in L^d$. I feel like I am just missing this last step but I am not sure on how to proceed.










    share|cite|improve this question

























      2












      2








      2







      $L_d = { phi in L_{[-a,a]}^2$ $vert$ $phi(t) = -phi(-t)$ $forall t in [-a,a] }$, prove that is a closed subspace of $L^2_{[-a,a]}$.



      Now, the subspace part is always the same and I managed.



      The part on closeness is where I stumble. This is what I did:



      I want to prove it using closeness by sequences, so:



      Let ${phi_n} subset L_d$ be a sequence, such that: $phi_n to phi$ in $L^2$. I want to prove that $phi in L_d$.



      I know that $phi_n to phi$ in $L^2$ implies that there exists an extracted subsequence $phi_{n_k}$ such that $phi_{n_k} to phi'$ almost everywhere. I also know that $phi' = phi$ almost everywhere for uniqueness of the limit. But still, I do not know how to say that $phi' = phi in L^d$. I feel like I am just missing this last step but I am not sure on how to proceed.










      share|cite|improve this question













      $L_d = { phi in L_{[-a,a]}^2$ $vert$ $phi(t) = -phi(-t)$ $forall t in [-a,a] }$, prove that is a closed subspace of $L^2_{[-a,a]}$.



      Now, the subspace part is always the same and I managed.



      The part on closeness is where I stumble. This is what I did:



      I want to prove it using closeness by sequences, so:



      Let ${phi_n} subset L_d$ be a sequence, such that: $phi_n to phi$ in $L^2$. I want to prove that $phi in L_d$.



      I know that $phi_n to phi$ in $L^2$ implies that there exists an extracted subsequence $phi_{n_k}$ such that $phi_{n_k} to phi'$ almost everywhere. I also know that $phi' = phi$ almost everywhere for uniqueness of the limit. But still, I do not know how to say that $phi' = phi in L^d$. I feel like I am just missing this last step but I am not sure on how to proceed.







      functional-analysis






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      asked Nov 27 '18 at 19:26









      qcc101

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      458113






















          1 Answer
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          Let $ phi^-(t) = phi(-t). $
          We want to show that $phi+phi^- = 0$.
          We have
          $$ Vert phi+phi^- Vert leq Vert phi-phi_n Vert + Vert phi_n+phi_n^- Vert + Vert phi^--phi_n^- Vert. $$
          The middle term is 0 since $phi_n in L_d$, and the other two terms converge to 0 by assumption.






          share|cite|improve this answer























          • I got your proof, but can you elaborate on the last step of mine?
            – qcc101
            Nov 27 '18 at 19:44










          • @qcc101 I'm not sure to be honest, but I don't think that the pointwise argument will work since you're assuming convergence in norm. See this post.
            – MisterRiemann
            Nov 27 '18 at 20:16











          Your Answer





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          active

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          1














          Let $ phi^-(t) = phi(-t). $
          We want to show that $phi+phi^- = 0$.
          We have
          $$ Vert phi+phi^- Vert leq Vert phi-phi_n Vert + Vert phi_n+phi_n^- Vert + Vert phi^--phi_n^- Vert. $$
          The middle term is 0 since $phi_n in L_d$, and the other two terms converge to 0 by assumption.






          share|cite|improve this answer























          • I got your proof, but can you elaborate on the last step of mine?
            – qcc101
            Nov 27 '18 at 19:44










          • @qcc101 I'm not sure to be honest, but I don't think that the pointwise argument will work since you're assuming convergence in norm. See this post.
            – MisterRiemann
            Nov 27 '18 at 20:16
















          1














          Let $ phi^-(t) = phi(-t). $
          We want to show that $phi+phi^- = 0$.
          We have
          $$ Vert phi+phi^- Vert leq Vert phi-phi_n Vert + Vert phi_n+phi_n^- Vert + Vert phi^--phi_n^- Vert. $$
          The middle term is 0 since $phi_n in L_d$, and the other two terms converge to 0 by assumption.






          share|cite|improve this answer























          • I got your proof, but can you elaborate on the last step of mine?
            – qcc101
            Nov 27 '18 at 19:44










          • @qcc101 I'm not sure to be honest, but I don't think that the pointwise argument will work since you're assuming convergence in norm. See this post.
            – MisterRiemann
            Nov 27 '18 at 20:16














          1












          1








          1






          Let $ phi^-(t) = phi(-t). $
          We want to show that $phi+phi^- = 0$.
          We have
          $$ Vert phi+phi^- Vert leq Vert phi-phi_n Vert + Vert phi_n+phi_n^- Vert + Vert phi^--phi_n^- Vert. $$
          The middle term is 0 since $phi_n in L_d$, and the other two terms converge to 0 by assumption.






          share|cite|improve this answer














          Let $ phi^-(t) = phi(-t). $
          We want to show that $phi+phi^- = 0$.
          We have
          $$ Vert phi+phi^- Vert leq Vert phi-phi_n Vert + Vert phi_n+phi_n^- Vert + Vert phi^--phi_n^- Vert. $$
          The middle term is 0 since $phi_n in L_d$, and the other two terms converge to 0 by assumption.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Nov 27 '18 at 20:12

























          answered Nov 27 '18 at 19:31









          MisterRiemann

          5,7791624




          5,7791624












          • I got your proof, but can you elaborate on the last step of mine?
            – qcc101
            Nov 27 '18 at 19:44










          • @qcc101 I'm not sure to be honest, but I don't think that the pointwise argument will work since you're assuming convergence in norm. See this post.
            – MisterRiemann
            Nov 27 '18 at 20:16


















          • I got your proof, but can you elaborate on the last step of mine?
            – qcc101
            Nov 27 '18 at 19:44










          • @qcc101 I'm not sure to be honest, but I don't think that the pointwise argument will work since you're assuming convergence in norm. See this post.
            – MisterRiemann
            Nov 27 '18 at 20:16
















          I got your proof, but can you elaborate on the last step of mine?
          – qcc101
          Nov 27 '18 at 19:44




          I got your proof, but can you elaborate on the last step of mine?
          – qcc101
          Nov 27 '18 at 19:44












          @qcc101 I'm not sure to be honest, but I don't think that the pointwise argument will work since you're assuming convergence in norm. See this post.
          – MisterRiemann
          Nov 27 '18 at 20:16




          @qcc101 I'm not sure to be honest, but I don't think that the pointwise argument will work since you're assuming convergence in norm. See this post.
          – MisterRiemann
          Nov 27 '18 at 20:16


















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