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- Show that C[0,1] is a vector space 2 answers
I'm a beginner in the course of Linear Algebra; please bear with me if the question seems too trivial.
The set of all continuous functions on interval $[0,1]$ is a vector space.
I have trouble in understanding this. What does a continuous function on $[0,1]$ mean? That the range lies within $[0,1]$?
For it to be a vector space, it needs to satisfy vector additivity.
Say, we take a vector with continuous function $f(t)=0.9$ which belongs to $V$. Another vector belonging to $V$ has continuous function $g(t)=0.8$.
For vector additivity, we add the elements of vector (in this case I'm considering only 1 element in the vector). Here the new vector would give us $0.8+0.9$ which us not in $[0,1]$. And yet this is a valid vector space.
I'm sure I'm missing something. I'm probably not able to understand what the question demands.