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Question

# The given set is a subset of C[−1, 1]. Which of these are also vector spaces? $F=\{f(x)$ in $C[1,1]: f(1)=0\}$

Solution

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Because the set is subspace of a vector space $C[-1,1]$, we only need to show $(c1), (c2), (a3), (a4)$.

Choose any $f,g\in F$ and any scalar $a$.

$(c1)$ $(f+g)(1)=f(1)+g(1)=0+0=0$. Conclude that $f+g \in F$.

$(c2)$ $(af)(1)=af(1)=a0=0$. Conclude that $af \in F$.

$(a3)$ Define $\theta=0$ where $0$ is nul-function. $0(1)=0$ so $0 \in F$. Also $(f+0)(t)=f(t)+0(t)=f(t)$.

$(a4)$ For given $f \in F$ define $-f$ as in $C[-1,1]$, $(-f)(t)=-f(t)$. $(-f)(1)=-f(1)=-0=0$ so $-f \in F$. $(f+(-f))(t)=f(t)+(-f)(t)=f(t)-f(t)=0$.

Conclude that $F$ is a vector space.

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