Direct sum problem
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
asked Nov 25 '18 at 11:01
user15269
1608
add a comment |
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
asked Nov 25 '18 at 11:01
user15269
1608
add a comment |
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
asked Nov 25 '18 at 11:01
user15269
1608
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
asked Nov 25 '18 at 11:01
user15269
1608
asked Nov 25 '18 at 11:01
user15269
1608
asked Nov 25 '18 at 11:01
user15269
1608
asked Nov 25 '18 at 11:01
user15269
1608
asked Nov 25 '18 at 11:01
user15269
1608
1608
add a comment |
add a comment |
1 Answer
1
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votes
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?
answered Nov 25 '18 at 11:29
egreg
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
add a comment |
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1 Answer
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active
oldest
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
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?
answered Nov 25 '18 at 11:29
egreg
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
add a comment |
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?
answered Nov 25 '18 at 11:29
egreg
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
add a comment |
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?
answered Nov 25 '18 at 11:29
egreg
178k1484201
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?
answered Nov 25 '18 at 11:29
egreg
178k1484201
answered Nov 25 '18 at 11:29
egreg
178k1484201
answered Nov 25 '18 at 11:29
egreg
178k1484201
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
add a comment |
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
add a comment |
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