Determine whether a set is subspace or not

In summary, the conversation discusses how to determine whether a set of functions is a subspace based on their integral values over a given interval. It is mentioned that for continuous functions, this can be easily checked by verifying closure conditions, but for integrable functions, the process is more complex. The individual suggests checking the conditions by taking two functions with zero integrals and ensuring that their sum also has a zero integral. They ask for help in solving this type of problem.
  • #1
mithila
1
0
The set of all functions f such that the integral of f(x) with respect to x over the interval [a,b] is
1. equal to zero
2. not equal to zero
3. equal to one
4. greater than equal to one
etc.

How can we determine this types of set is a subspace of not.



for the case of the set of continuous function we can easily check the closure conditions but for integrable function how can we check that.

can anyone help me to solve this type of problem

Thanks.
 
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  • #2
Usually, you just check the conditions. For example, take two functions f and g whose integral over [a, b] is zero, and check that the integral of (f + g) over [a, b] is also zero, etc.
 

FAQ: Determine whether a set is subspace or not

What is a subspace?

A subspace is a subset of a vector space that satisfies all the properties of a vector space, such as closure under addition and scalar multiplication.

How do you determine if a set is a subspace?

To determine if a set is a subspace, you need to check if it satisfies all the properties of a vector space. This includes checking if the set is closed under addition and scalar multiplication, contains the zero vector, and if it follows the associative and distributive properties.

What is the difference between a subspace and a vector space?

A subspace is a subset of a vector space that satisfies all the properties of a vector space, while a vector space is a set of vectors that follow specific rules and properties. A subspace is always a vector space, but a vector space may not necessarily be a subspace.

Can a set be a subspace of multiple vector spaces?

Yes, a set can be a subspace of multiple vector spaces as long as it satisfies the properties of a vector space for each vector space it is a part of.

What are some common mistakes when determining if a set is a subspace?

Some common mistakes when determining if a set is a subspace include not checking all the properties of a vector space, assuming that a set is a subspace just because it contains vectors, and not considering the specific vector space that the set is a part of.

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