Functions forming a vector space

[Pushoam]

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?

 

[PeroK]

Technically, the zero function is periodic for any given period. In particular, it has the same value at the end points.

 

[Ray Vickson]

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

View attachment 229213

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?

You need to have confidence in your own reasoning. (i) Have you verified all the defining properties of vector spaces in (1)-(2)? (ii) Have you shown that some vector-space property fails in (3)?

 

[Mark44]

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

View attachment 229213

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?

What's the definition of 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?

 

[fresh_42]

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.

The notion of the corresponding equation would have been almost shorter: $(\alpha f + \beta g)(0)=\alpha f(0)+\beta g(0)=\alpha f(L)+ \beta g(L)=(\alpha f + \beta g)(L)$

2) Similarly, for 2 also all properties get verified except existence of a null vector.

Why this? $0(0)=0=0(L)$

Can a function h(x) = 0 for all x, be a periodic function?

This is the extreme version of a periodic function, namely with period $0$. There is no reason to exclude it.

3) For 3, this set of functions do not have closure feature. So, it does not form a vector space.

Yes, and again $0(0)=0\neq 4$

Is this correct?

Yes, except that I couldn't see your reasoning why the null vector in section 2 shouldn't work.

Edit: Please do the following little exercise: A set which is closed under addition and scalar multiplication automatically contains zero.

 

[Pushoam]

You need to have confidence in your own reasoning. (i) Have you verified all the defining properties of vector spaces in (1)-(2)? (ii) Have you shown that some vector-space property fails in (3)?

Yes, I verified it in mind. I didn't write the verification here as it was easy.

 

[Mark44]

This is the extreme version of a periodic function, namely with period 0.

Or any period.
If f(x) = 0 for all x, then f(x) = f(x + L), for any L.

The vector space properties in the attached image in post 1 is a bit unusual. The usual definition includes statements about closure under vector addition and closure under scalar multiplication.

Minor point: The usual definitions are of the properties these operations. I've never seen them called features.

 

[vela]

This is the extreme version of a periodic function, namely with period $0$. There is no reason to exclude it.

Typically the period is required to be positive to exclude possibilities like claiming f(x)=x is periodic since f(x)=f(x+0) for all x.

 

[fresh_42]

Typically the period is required to be positive to exclude possibilities like claiming f(x)=x is periodic since f(x)=f(x+0) for all x.

Makes sense. I thought the zero might be necessary for the set of all periods, or to get vector spaces for periodic functions, but "any" sounds better than "0" although "any" includes it.

 

[Pushoam]

The notion of the corresponding equation would have been almost shorter: $(\alpha f + \beta g)(0)=\alpha f(0)+\beta g(0)=\alpha f(L)+ \beta g(L)=(\alpha f + \beta g)(L)$Edit: Please do the following little exercise: A set which is closed under addition and scalar multiplication automatically contains zero.

Let's say that $\alpha$ and $\beta$ belongs to this set and a, b belong to the scalar field. Then, the set being closed under addition and scalar multiplication means that $a \alpha + b \beta$ must belong to the set.
Taking $b \beta = - a \alpha$ implies that 0 belongs to the set.

Thus, a set which is closed under addition and scalar multiplication automatically contains zero.