Continuous Functions, Vector Spaces

In summary, the set of all continuous functions on the interval (a,b) of the real line is likely a vector space, as the sum of two continuous functions is also continuous. However, there may be some issues with infinite series in the definition that could potentially impact this conclusion.
  • #1
psholtz
136
0

Homework Statement


Is the set of all continuous functions (defined on say, the interval (a,b) of the real line) a vector space?


Homework Equations


None.


The Attempt at a Solution


I'm inclined to say "yes", since if I have two continuous functions, say, f and g, then their sum f+g is also continuous.

But as per Fourier analysis, one can "arrive" at discontinuous functions by taking an infinite series of continuous functions.

Soo.. I'm not 100% certain.
 
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  • #2
I believe it is.
I don't think infinite sums are "allowed" in the definition; if they were then there would be convergence issues and certain things that ought to be vector spaces would not be.

I believe the definition require the sum of a finite linear combination to be in the vector space.
 

FAQ: Continuous Functions, Vector Spaces

What is the definition of a continuous function?

A continuous function is a mathematical function that has no sudden or discontinuous changes. This means that the graph of the function can be drawn without lifting the pen from the paper.

What is the difference between pointwise and uniform convergence of a sequence of continuous functions?

Pointwise convergence means that for each point in the domain, the sequence of function values converges to the corresponding value of the limit function. Uniform convergence means that for any small interval, the sequence of function values eventually gets closer and closer to the values of the limit function.

What is a vector space and what are its main properties?

A vector space is a set of objects called vectors, which can be added together and multiplied by numbers, called scalars. Its main properties include closure under addition and scalar multiplication, associativity, commutativity, distributivity, and the existence of an identity element and inverse elements.

How do you prove that a set is a vector space?

To prove that a set is a vector space, you must show that it satisfies all of the main properties of a vector space, such as closure under addition and scalar multiplication. You can do this by demonstrating that the operations of addition and scalar multiplication are well-defined, and that the set contains an identity element and inverse elements.

What is the dimension of a vector space and how is it determined?

The dimension of a vector space is the number of vectors in a basis for that space. It is determined by finding the minimum number of linearly independent vectors that span the entire vector space. This can be done by reducing the matrix of the vectors to its reduced row echelon form and counting the number of nonzero rows.

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