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1. The problem statement, all variables and given/known data
1.1.3
1) Do functions that vanish at the endpoints x=0 and L=0 form a vector space?
2) How about periodic functions? obeying f(0)=f(L) ?
3) How about functions that obey f(0)=4 ?
If the functions do not qualify, list what go wrong.
2. Relevant equations
3. The attempt at a solution
1) Considering a set of functions which vanish at the end points x = {0,L}. Let's say f,g,h belong to this set of functions. Then all the properties in the above image are verified. So, this set of functions form a vector space.
2) Similarly, for 2 also all properties get verified except existence of a null vector. Can a function h(x) = 0 for all x, be a periodic function?
3) For 3, this set of functions do not have closure feature. So, it does not form a vector space.
Is this correct?
1.1.3
1) Do functions that vanish at the endpoints x=0 and L=0 form a vector space?
2) How about periodic functions? obeying f(0)=f(L) ?
3) How about functions that obey f(0)=4 ?
If the functions do not qualify, list what go wrong.
2. Relevant equations
3. The attempt at a solution
1) Considering a set of functions which vanish at the end points x = {0,L}. Let's say f,g,h belong to this set of functions. Then all the properties in the above image are verified. So, this set of functions form a vector space.
2) Similarly, for 2 also all properties get verified except existence of a null vector. Can a function h(x) = 0 for all x, be a periodic function?
3) For 3, this set of functions do not have closure feature. So, it does not form a vector space.
Is this correct?
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