Caracterizing a subspace of L^2

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In summary, the conversation discusses the subspace M of L²([0,1]) consisting of functions with vanishing mean and the dimension of its orthogonal subspace M^{\perp}. It is suggested that M^{\perp} is likely to consist of constant functions, but a proof has not been found yet. The conversation ends with one person studying for finals and the other expressing appreciation for talking to an intelligent individual.
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quasar987
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[SOLVED] Caracterizing a subspace of L^2

Homework Statement



Call M the subspace of L²([0,1]) consisting of all functions of vanishing mean. I.e., [itex]u\in M \Rightarrow \int_0^1u(s)ds=0[/itex].

I am trying to find the dimension of the orthogonal of M,

[tex]M^{\perp}=\{x\in L^2([0,1]):\int_0^1x(s)u(s)ds=0 \ \forall u\in M\}[/tex]

I would be surprised if [itex]M^{\perp}[/itex] was anything other than the constant functions, but my attempts at a proof have been unsuccessful.

Any idea how to go at this?
 
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  • #2
Solved.
 
Last edited:
  • #3
Was it the (a.e.) constant functions in the end?

Sorry I couldn't help -- been busy with studying for my finals!
 
  • #4
morphism said:
Was it the (a.e.) constant functions in the end?

Sorry I couldn't help -- been busy with studying for my finals!

How could it be anything else? The set of vanishing mean functions is basically defined as the set of all u(x) such that <u(x),1>=0. L^2 is a norm up to a.e.
 
  • #5
Well, yeah... :tongue2:
 
  • #6
morphism said:
Well, yeah... :tongue2:

Nice to talk to an erudite and sophisticated gentleman for a change. Instead of a bonehead. :)
 

1. What is a subspace of L^2?

A subspace of L^2 is a subset of the Hilbert space L^2, which is a mathematical space used to describe functions that have a finite energy. In other words, it is a space of square-integrable functions.

2. How is a subspace of L^2 characterized?

A subspace of L^2 is characterized by its basis, which is a set of linearly independent functions that span the subspace. These functions must also be square-integrable and have a finite energy.

3. What are the properties of a subspace of L^2?

A subspace of L^2 has the following properties: it is closed under addition and scalar multiplication, it contains the zero vector, and it is a vector space. Additionally, it is a Hilbert space, meaning it has an inner product defined on it.

4. What is the importance of characterizing a subspace of L^2?

Characterizing a subspace of L^2 allows for a better understanding and analysis of the functions within that space. It also allows for the application of various mathematical techniques and tools, such as orthogonal projections and Fourier series, to study the functions in the subspace.

5. How is a subspace of L^2 different from L^2?

A subspace of L^2 is a subset of L^2, meaning it contains only a specific type of functions within the larger space. L^2, on the other hand, contains all square-integrable functions. Additionally, a subspace of L^2 has a defined basis, while L^2 does not necessarily have a basis.

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