How can you compare equality of two vectors of infinite dimension? E.g., say we have 2 vectors of infinite dimension: $$A=&lt;1,0,1,1,...,1&gt;$$$$B=&lt;0,1,1,1,...,1&gt;$$
These are qualitatively not the same vector, though it seems hard to definitively prove that.
If you define equality as $A_i=B_i$ for all elements $i$, then it seems to follow that $A\neq B$.
However, if you define equality between vectors as $\Vert A \Vert =\Vert B \Vert$ and $A$ being co-linear and co-directed with $B$, then these vectors might seem equal, as $\Vert A \Vert = \Vert B \Vert = ∞$, and the angle between the two vectors converges upon $0$ at $∞$ (although by the formula $cos \theta=\frac{&lt;A, B&gt;}{\Vert A \Vert \Vert B \Vert}$, this might technically be indeterminate, I suppose).
Is there a better way to understand and rationalize the relationship between infinite dimension vectors?
Let's use a more concrete example to demonstrate this point, which you are probably already quite familiar with.
Consider all the continuous functions defined on the interval $[0,1]$. This is usually notated as $C[0,1]$. Firstly, is this a vector space? The answer is yes
What is the dimension of this space? It is infinite - as you need to define the value of the function at each point in $[0,1]$ - of which there are infinitely many
Now, when are two vectors (functions) from this space equal? When their pointwise match is equal - or in more mathematical terms
$$f = g \space \text{iff} \space \space ||f-g|| = 0$$
where || is a norm function defined on the vector space
Let's use a more concrete example to demonstrate this point, which you are probably already quite familiar with.
Consider all the continuous functions defined on the interval $[0,1]$. This is usually notated as $C[0,1]$. Firstly, is this a vector space? The answer is yes
What is the dimension of this space? It is infinite - as you need to define the value of the function at each point in $[0,1]$ - of which there are infinitely many
Now, when are two vectors (functions) from this space equal? When their pointwise match is equal - or in more mathematical terms
$$f = g \space \text{iff} \space \space ||f-g|| = 0$$
where || is a norm function defined on the vector space