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I have a question from my functional analysis course I am a little confused with; here goes:

Let $X = C[-1,1]$ be the linear space of continuous functions on the interval $[-1,1]$ over the field $\mathbb{R}$.

Consider the subset $M$ $=$ {${f(x) \in C[-1,1] : f(-1) = f(1)}$}

Is $M$ a linear subspace of $X?$

I think my problem lies in not fully grasping the original vector space or in fact thinking it to be more complicated than it already is because I am happy with the usual definitions of vector spaces and subspaces.

Obviously we have a zero function, namely the constant function always equal to zero, which also satisfies the additional criteria that $f(-1) = f(1)$ as both equal zero.

From calculus we know that if $f$ and $g$ are continuous functions and $\alpha$ a constant then both $f + g$ and $\alpha f$ are continuous functions, however how do I know if $(f+g)(-1) = (f+g)(1)$? Can one simply use the argument that $(f+g)(-1) = f(-1) + g(-1) = f(1) + g(1) = (f+g)(1)$?

I assume $\alpha$$f(-1)$ will equal $\alpha f(1)$ because we are multiplying both sides by the same constant.

Any help would be much appreciated, thank you in advance!

katherinebarry
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1 Answers1

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Since $f,g\in M$ you know that $f(-1)=f(1)$ and $g(-1)=g(1)$. Therefore, adding them together yields $\underbrace{\underbrace{f(-1)+g(-1)}_{\displaystyle (f+g)(-1)}=\underbrace{f(1)+g(1)}_{\displaystyle (f+g)(1)}}_{\Longrightarrow \displaystyle (f+g)(-1)=(f+g)(1)}$.

Everything you did is correct and you're done.

Git Gud
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