Are Continuous Functions with Zero Integral a Subspace of C[a,b]?

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SUMMARY

The set of continuous functions \( f = f(x) \) on the interval \([a,b]\) such that \(\int_a^b f(x) \, dx = 0\) is indeed a subspace of \( C[a,b] \). This conclusion is based on the definition of a subspace, which requires closure under addition and scalar multiplication. Specifically, if \(\int_a^b f(x) \, dx = 0\) and \(\int_a^b g(x) \, dx = 0\), then it follows that \(\int_a^b (f(x) + g(x)) \, dx = 0\). Additionally, for any scalar \( k \), \(\int_a^b k f(x) \, dx = k \int_a^b f(x) \, dx = 0\), confirming the closure properties.

PREREQUISITES
  • Understanding of continuous functions in the context of real analysis
  • Familiarity with the properties of integrals, specifically definite integrals
  • Knowledge of vector space concepts, particularly subspaces
  • Basic proficiency in mathematical notation and operations
NEXT STEPS
  • Review the definition and properties of vector spaces and subspaces
  • Study the properties of integrals, focusing on linearity and closure
  • Explore examples of other subspaces within \( C[a,b] \)
  • Investigate the implications of the zero integral condition in functional analysis
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Mathematics students, particularly those studying real analysis or functional analysis, as well as educators seeking to clarify the concept of subspaces in the context of continuous functions.

gaborfk
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Yet another problem I need to get some starting help on:

Show that the set of continuous functions f=f(x) on [a,b] such that \int \limits_a^b f(x) dx=0 is a subspace of C[a,b]
Thank you
 
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I would start by checking the definition of subspace.
 
Definition of subspace means that the functions are closed under addition and scalar multiplication
 
gaborfk said:
Definition of subspace means that the functions are closed under addition and scalar multiplication

So can you show that that's true for the potential subspace in your example?
 
You mean that if \int \limits_a^b f(x) dx=0 and \int \limits_a^b g(x) dx=0, can I prove that \int \limits_a^b f(x)+g(x) dx=0? Also, if \int \limits_a^b f(x) dx=0 then k\int \limits_a^b f(x) dx=0?
 
Yeah, that's pretty much it. (Technically you also have to show that it's a subset, but in this case that's trivial.)
 
Thank you!

The "hard ones" are so easy sometimes...
 

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