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My question is are there subsets $A$ and $B$whereby $A+B$ is a subspace if $A$ and $B$ aren't subspaces?

I know that $A+B$ is a subspace if $A$ and $B$ are and can prove it by showing it's closed under addition and scalar multiplication.

However, if $A$ and $B$ aren't subspaces I think I'd have to prove that $A+B$ isn't a subspace by going through every property of them not being a subspace, e.g if $A$ isn't a subspace then it doesn't contain the zero vector so $A+B$ can't be a subspace, then do if $A$ isn't a subspace it's not closed under addition so $A+B$ can't be etc...

I feel like this is not a very efficient proof and was wondering if there are any other ways of doing it? Sorry for the long description.

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    Sure. $A=\{(x,y) : x \ge 0\}$ and $B=\{(x,y) : x < 0\}.$ Then the formal sum $A+B$ is equal to $\mathbf R^2.$ – 2017-02-11
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    What happens if $B$ is a subspace and $A$ has just two elements, $0$ and some other element of $B$? – 2017-02-11

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Try a line that passes through the origin and cut it in two parts.

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    If i then wanted to prove this also for A-B how would that work? – 2017-02-11
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    How do you define $A - B$? – 2017-02-11
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    Cut the line in two parts $A$ and $C$, then let $B = -C.$ Neither $A$ or $B$ is a subspace but $A-B$ is the original line passing through the origin. – 2017-02-11