Direct sum problem












1














In vector space $mathbb{R} ^ mathbb{R} =left{ f mid f colon mathbb{R} to mathbb{R} right} $ let L be a set of even functions $( f(-t)=f(t))$ and and M a set of odd functions
$( f(-t)=-f(t))$ .It's easy to prove that L and M are subspaces of the vector space but how do I prove that
$mathbb{R} ^ mathbb{R} = M oplus L$ ? It's easy to prove that nul function is in the intersection of M and L but how can a function that isn't even or odd be a sum of two functions that are even or odd. For example the exponential function $ e^x$ ?






vector-spaces







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    1














    In vector space $mathbb{R} ^ mathbb{R} =left{ f mid f colon mathbb{R} to mathbb{R} right} $ let L be a set of even functions $( f(-t)=f(t))$ and and M a set of odd functions
    $( f(-t)=-f(t))$ .It's easy to prove that L and M are subspaces of the vector space but how do I prove that
    $mathbb{R} ^ mathbb{R} = M oplus L$ ? It's easy to prove that nul function is in the intersection of M and L but how can a function that isn't even or odd be a sum of two functions that are even or odd. For example the exponential function $ e^x$ ?






    vector-spaces







    share|cite|improve this question

























      1












      1








      1







      In vector space $mathbb{R} ^ mathbb{R} =left{ f mid f colon mathbb{R} to mathbb{R} right} $ let L be a set of even functions $( f(-t)=f(t))$ and and M a set of odd functions
      $( f(-t)=-f(t))$ .It's easy to prove that L and M are subspaces of the vector space but how do I prove that
      $mathbb{R} ^ mathbb{R} = M oplus L$ ? It's easy to prove that nul function is in the intersection of M and L but how can a function that isn't even or odd be a sum of two functions that are even or odd. For example the exponential function $ e^x$ ?






      vector-spaces







      share|cite|improve this question













      In vector space $mathbb{R} ^ mathbb{R} =left{ f mid f colon mathbb{R} to mathbb{R} right} $ let L be a set of even functions $( f(-t)=f(t))$ and and M a set of odd functions
      $( f(-t)=-f(t))$ .It's easy to prove that L and M are subspaces of the vector space but how do I prove that
      $mathbb{R} ^ mathbb{R} = M oplus L$ ? It's easy to prove that nul function is in the intersection of M and L but how can a function that isn't even or odd be a sum of two functions that are even or odd. For example the exponential function $ e^x$ ?






      vector-spaces




      vector-spaces






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      share|cite|improve this question











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      share|cite|improve this question










      asked Nov 25 '18 at 11:01









      user15269

      1608




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          Suppose you are able to write $f$ as the sum of an even function $g$ and an odd function $h$; then, for every $tinmathbb{R}$,
          begin{align}
          f(t)&=g(t)+h(t)\
          f(-t)&=g(-t)+h(-t)=g(t)-h(t)
          end{align}
          Can you go on?






          share|cite|improve this answer





















          • Does this mean that not every function can be written in this form? For example $ e^x + frac{1}{e^x} $ would then have to be an even function which it isn't.
            – user15269
            Nov 25 '18 at 11:44










          • @user15269 No, every function can be written as a sum of an even and an odd function; if the function is even to begin with, the “odd part” is the constant zero function (which is odd, isn't it?)
            – egreg
            Nov 25 '18 at 11:51










          • Thanks I get it
            – user15269
            Nov 25 '18 at 12:07













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          0














          Suppose you are able to write $f$ as the sum of an even function $g$ and an odd function $h$; then, for every $tinmathbb{R}$,
          begin{align}
          f(t)&=g(t)+h(t)\
          f(-t)&=g(-t)+h(-t)=g(t)-h(t)
          end{align}
          Can you go on?






          share|cite|improve this answer





















          • Does this mean that not every function can be written in this form? For example $ e^x + frac{1}{e^x} $ would then have to be an even function which it isn't.
            – user15269
            Nov 25 '18 at 11:44










          • @user15269 No, every function can be written as a sum of an even and an odd function; if the function is even to begin with, the “odd part” is the constant zero function (which is odd, isn't it?)
            – egreg
            Nov 25 '18 at 11:51










          • Thanks I get it
            – user15269
            Nov 25 '18 at 12:07


















          0














          Suppose you are able to write $f$ as the sum of an even function $g$ and an odd function $h$; then, for every $tinmathbb{R}$,
          begin{align}
          f(t)&=g(t)+h(t)\
          f(-t)&=g(-t)+h(-t)=g(t)-h(t)
          end{align}
          Can you go on?






          share|cite|improve this answer





















          • Does this mean that not every function can be written in this form? For example $ e^x + frac{1}{e^x} $ would then have to be an even function which it isn't.
            – user15269
            Nov 25 '18 at 11:44










          • @user15269 No, every function can be written as a sum of an even and an odd function; if the function is even to begin with, the “odd part” is the constant zero function (which is odd, isn't it?)
            – egreg
            Nov 25 '18 at 11:51










          • Thanks I get it
            – user15269
            Nov 25 '18 at 12:07
















          0












          0








          0






          Suppose you are able to write $f$ as the sum of an even function $g$ and an odd function $h$; then, for every $tinmathbb{R}$,
          begin{align}
          f(t)&=g(t)+h(t)\
          f(-t)&=g(-t)+h(-t)=g(t)-h(t)
          end{align}
          Can you go on?






          share|cite|improve this answer












          Suppose you are able to write $f$ as the sum of an even function $g$ and an odd function $h$; then, for every $tinmathbb{R}$,
          begin{align}
          f(t)&=g(t)+h(t)\
          f(-t)&=g(-t)+h(-t)=g(t)-h(t)
          end{align}
          Can you go on?







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Nov 25 '18 at 11:29









          egreg

          178k1484201




          178k1484201












          • Does this mean that not every function can be written in this form? For example $ e^x + frac{1}{e^x} $ would then have to be an even function which it isn't.
            – user15269
            Nov 25 '18 at 11:44










          • @user15269 No, every function can be written as a sum of an even and an odd function; if the function is even to begin with, the “odd part” is the constant zero function (which is odd, isn't it?)
            – egreg
            Nov 25 '18 at 11:51










          • Thanks I get it
            – user15269
            Nov 25 '18 at 12:07




















          • Does this mean that not every function can be written in this form? For example $ e^x + frac{1}{e^x} $ would then have to be an even function which it isn't.
            – user15269
            Nov 25 '18 at 11:44










          • @user15269 No, every function can be written as a sum of an even and an odd function; if the function is even to begin with, the “odd part” is the constant zero function (which is odd, isn't it?)
            – egreg
            Nov 25 '18 at 11:51










          • Thanks I get it
            – user15269
            Nov 25 '18 at 12:07


















          Does this mean that not every function can be written in this form? For example $ e^x + frac{1}{e^x} $ would then have to be an even function which it isn't.
          – user15269
          Nov 25 '18 at 11:44




          Does this mean that not every function can be written in this form? For example $ e^x + frac{1}{e^x} $ would then have to be an even function which it isn't.
          – user15269
          Nov 25 '18 at 11:44












          @user15269 No, every function can be written as a sum of an even and an odd function; if the function is even to begin with, the “odd part” is the constant zero function (which is odd, isn't it?)
          – egreg
          Nov 25 '18 at 11:51




          @user15269 No, every function can be written as a sum of an even and an odd function; if the function is even to begin with, the “odd part” is the constant zero function (which is odd, isn't it?)
          – egreg
          Nov 25 '18 at 11:51












          Thanks I get it
          – user15269
          Nov 25 '18 at 12:07






          Thanks I get it
          – user15269
          Nov 25 '18 at 12:07




















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