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Usually, the zero vector in a vector space can be found quite easily. Often it's just the intuitive sense of 'zero' in the set that the vector space is defined over.

For example, if we consider the vector space of real continuous functions in an interval, then the zero vector is simply the zero function.

If we consider the vector space of complex numbers, the zero vector is simply $0+0i$.

What is an example of a vector space where the zero vector is non-trivial?

Obviously the idea of non-triviality is somewhat subjective, but that is very much unavoidable for a question of this nature.

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If you look in $L^p$ spaces, the zero-vector is an equivalence class of functions, which might be non-trivial in your sense. For example, in $L^2({\mathbb{R}},\mu)$, where $\mu$ is the Lebesgue measure, the function $$f(x) = 1_{\mathbb{Q}}(x) = \begin{cases} 1 & x \in \mathbb{Q} \\ 0 & \text{otherwise} \end{cases}$$ is a representative of the $0$ function class (i.e., a representative of the zero-vector).

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Consider $\Bbb{R}_+$ over $\Bbb{R}$ with $(a)+(b)=(ab)$ and $c(a)=(a^c)$. Then $\vec{0}=(1)$.

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