Question on the proof that the set of martingale $M$ s.t. $\mathbb E M_1^2<\infty $ is a normed space. (but its linear algebra)

Question

Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space with the filtration $(\mathcal F_t)_t$. Let $H^2$ the set of the continuous martingale $M=(M_t)_{t\in [0,1]}$ with the norm $\|M\|_{H^2}=\sqrt{\mathbb EM_1^2}$. The question is to show that $(H^2,\|\cdot \|_{H^2})$ is a normed space. I have no problem to show that $H^2$ is a vector space. Now, in my course they show that if $M,N\in H^2$ then for all $\alpha ,\beta \in \mathbb R$, $$\|\alpha M+\beta N\|_{H^2}<\infty, $$ but I don't understand why we do this last step. Is it really necessary ? And if yes, could you please give me an example where $V$ is a vector space, $\|\cdot \|_V$ is a norm on $V$, but there is $\alpha ,\beta \in\mathbb R$ and $v,w\in V$ such that $\|\alpha v+\beta w\|_V=\infty $ ? Because this last verification looks to be a tautology to me. Indeed, if $\alpha ,\beta \in\mathbb R$, $v,w\in V$, then $$\|\alpha u+\beta v\|\leq |\alpha |\underbrace{\|u\|_V}_{<\infty }+|\beta |\underbrace{\|v\|_V}_{<\infty }<\infty $$

Answer 1

Yes it really is necessary. Why ? Well the definition of $H^{2}$ is that it is the set of continous martingales with $||M|| < \infty$ So you d have to prove the following things: (1) is $\beta M_t + \alpha N_t $ continous? This is easy.(2) Is it a martingale ? This also follows easily as the conditional expectation is linear. (3) is the norm of $\beta M_t + \alpha N_t $ finite ? Yes this is also the case by triangular inequality. The fact that the axioms of a vector space are fulfilled then follows as $M_t+N_t$ is defined by pointwise addition.

Cheers