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If I let $V=c([a,b])$ be the vector space consisting of all functions $f(t)$ which are defined and continuous on the interval $0\le t\le1$, what are some conditions that define subspaces of $V$?

For $f(1-t) = -tf(t)$ to be considered a subspace of $V$ I got that $h(1-t)$ such that $-th(t) = f(1-t) + g(1-t) = -tf(t) - tg(t)$, which is a subspace and pretty straightforward, but how will I approach this particular condition, $f(0) =2f(1)$ or even $f(0)f(1)=1$?

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In general, a subset $W$ of a vector space $V$ is a subspace of $V$ if and only if for all scalars $\lambda, \mu$ and all vectors $v,w \in W$ we have $\lambda v + \mu w \in W$.

So, for instance, if $W$ is the subset of $V = \mathcal{C}[a,b]$ consisting of all functions that satisfy $f(1-t)=-tf(t)$, then $W$ is a subspace if and only if for all real numbers $\lambda, \mu \in \mathbb{R}$ we have $\lambda f(1-t) + \mu g(1-t) = -\lambda tf(t) - \mu tg(t)$ Is this condition satisfied?

If in addition we introduce the constraint that each $f \in W$ satisfies $f(0)=2f(1)$, or respectively $f(0)f(1)=1$, then we require: $\lambda f(0) + \mu g(0) = 2\lambda f(1) + 2\mu g(1)$ or respectively $(\lambda f(0) + \mu g(0))(\lambda f(1) + \mu g(1)) = 1$ Do these conditions hold? An affirmative or negative answer to this question will tell you whether or not you have a subspace.

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    :) thank you very much!!2012-09-24