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## Homework Statement

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.

## Homework Equations

## 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?