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Let $ ‎X=C[-1,1]‎$ be‎‎ and $M, N $ are the subspaces defined by ‎$$ ‎M= ‎‎\left\{f‎ \in ‎X\mid ‎f(t)=0 ,‎ ‎t \in [-1, 0] \right\}, ‎$$ $$ ‎N= ‎‎\left\{g \in ‎X\mid ‎g(t)=0 ,‎ ‎‎t \in [0,1]\right\}.$$ What are $ M \cap N \text{ and }M + N$?

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When you write $M+N$, in what context do you mean that? – Asaf Karagila Mar 28 '13 at 10:04
I want to see if $ M \bigoplus N =X ?$ – nim Mar 28 '13 at 10:10
To see that simply take any function in $X$ and see if you can write it as a sum of two functions from $M$ and from $N$. Can you do that for the function $f(t)=1$? – Asaf Karagila Mar 28 '13 at 10:12
so $ M \bigoplus N \neq X$ ? thanks – nim Mar 28 '13 at 10:16
up vote 1 down vote accepted

Hint: If $h\in M\cap N$ then for every $t\in[-1,0]$, $h(t)=0$ and for every $t\in[0,1]$, $h(t)=0$.

Hint II: Show that if $f\in M+N$ then $f(0)=0$, and conclude that $M\oplus N\neq X$.

Hint III: Show that if $h(0)=0$ then you can write $h$ as the sum of $f\in M$ and $g\in N$.

Conclude what is $M\oplus N$ from the second and third hints. They give sufficient and necessary conditions for a function to be in the sum.

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And one can even determine $M+N$... – Did Mar 28 '13 at 23:34
Did: Please try to determine $ M+N= ?$ – nim Mar 29 '13 at 1:40
@nim: Only the zero function has that for every $t$, $h(t)=0$. So the sum of two non-zero spaces are clearly not just the zero function. – Asaf Karagila Mar 30 '13 at 7:55
@nim: I already told you almost everything. If you really think that you can handle a Ph.D. in mathematics then you should be able to figure it out from the hints in my answer above. – Asaf Karagila Mar 30 '13 at 8:33
@nim: No, it seems that not everything is clear. Because if it were, the answer would have been obvious. – Asaf Karagila Mar 30 '13 at 9:38

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