sequence space and vector space

Question

A property of a vector space V is this:

$\forall x \in V:$ $0\cdot x=\vec{0}$

Now, consider the space of sequences $\left ( x_{1},\cdot \cdot \cdot ,x_{i},\cdot \cdot \cdot \right )$ with $x_{i} \in \mathbb{R}$.

Applying the above property of a vector space,

we get $0\cdot \left ( x_{n} \right )_{n \in \mathbb{N}}=0\cdot \left ( x_{1},\cdot \cdot \cdot ,x_{n} \right )=\left ( 0 \cdot x_{1}, \cdot \cdot \cdot , 0\cdot x_{n}\right )=\left ( 0,\cdot \cdot \cdot ,0 \right )$.

But how is this $\vec{0}$?

Thanks in advance.

Answer 1

Try to remember what a vector space is: it is an abelian group together with a field and some properties attached to it.

As such, an abelian group has a unit, and units in abelina groups are typically denoted as $0$ (and the operation as $+$)

The symbol $\vec 0$ means symply "the zero of the Abelian group that is $V$). In your case, $V$ is the space of all sequences of real numbers, i.e.

$$V=\{(x_1,x_2,x_3,\dots)| \forall i\in\mathbb N: x_i\in\mathbb R\}$$

and, if no other detail is given, the operation is probably (but please check to see if that is true)

$$(x_1, x_2,\dots) + (y_1,y_2,\dots) = (x_1+y_1, x_2+y_2,\dots)$$

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Now, you need to realize what the $0$ element is. By definition, it is

That element $\vec e\in V$ for which $x+\vec e=x$ for all elemenets $x\in V$.

It should now be very clear which element you are looking for.