Meaning of a Subspace

  • #1
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Hi PF!

I want to make sure I understand the notion of a subspace. Our professor gave an example of one: the set of degree n polynomials is a subspace of continuous functions. This is because a) a polynomial is intrinsically a subset of continuous functions and b) summing any polynomials yields another polynomial (closed under addition) and c) any polynomial multiplied by a constant is still a polynomial (closed under scalar multiplication).

If my reasoning is correct, then I think the set of discontinuous functions is NOT a subset of all real functions. To see why, consider ##f(x) = 1## everywhere except non-existent at ##x=1##. Then take the function ##g(x)=1## when ##x=1## and non-existent everywhere else. Then the sum is clearly continuous. (How would this change though if I changed non-existent to zero? I know the result is the same, but is ##g(x)## as I have defined it even discontinuous--I realize this is a real analysis question.)
 

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  • #2
PeroK
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Hi PF!

I want to make sure I understand the notion of a subspace. Our professor gave an example of one: the set of degree n polynomials is a subspace of continuous functions. This is because a) a polynomial is intrinsically a subset of continuous functions and b) summing any polynomials yields another polynomial (closed under addition) and c) any polynomial multiplied by a constant is still a polynomial (closed under scalar multiplication).

If my reasoning is correct, then I think the set of discontinuous functions is NOT a subset of all real functions. To see why, consider ##f(x) = 1## everywhere except non-existent at ##x=1##. Then take the function ##g(x)=1## when ##x=1## and non-existent everywhere else. Then the sum is clearly continuous. (How would this change though if I changed non-existent to zero? I know the result is the same, but is ##g(x)## as I have defined it even discontinuous--I realize this is a real analysis question.)
You can't choose functions that are not defined at some point, as functions that have different domains can't be added so they don't form a vector space. The space of all real-valued functions would imply that they all all defined on some fixed domain.

Also, functions that are not defined at a point are not necessarily discontinuous. They may be continuous on their domain. A good example is the function ##1/x##, which is a continuous function on its domain.

Instead, you need to think of two genuinely discontinuous functions that can be added to form a contiuous function.

Or, perhaps you could think of a simpler example using a subset of the polynomials that does not form a subspace?
 
  • #3
S.G. Janssens
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Our professor gave an example of one: the set of degree n polynomials is a subspace of continuous functions.
I would say: "The set of polynomials of degree at most ##n##". (Here you regard the zero polynomial as having degree ##-\infty## or you should stipulate that this set includes the zero polynomial.)
If my reasoning is correct, then I think the set of discontinuous functions is NOT a subset of all real functions.
You mean to say "subspace" instead of "subset"? In addition, I very much second the remarks made by PeroK. They appeared on my screen while I was writing this.
Instead, you need to think of two genuinely discontinuous functions that can be added to form a contiuous function.
Or take a discontinuous function and multiply it by zero.
 
  • #4
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Thank you both for finessing my logic! I really appreciate both of your input!:biggrin:
 

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