Double Orthogonal Closed Subspace Inner Product => Hilbert

• SqueeSpleen
In summary, the conversation discusses the conditions for X to be a Hilbert Space, which is a complete Inner Product Space. It is stated that if for every closed subspace M, the orthogonal complement of the orthogonal complement of M is equal to M, then X is a Hilbert Space. The hint suggests using a map T to prove this statement. The author attempts to solve the problem by proving that the closure of the image of T on the orthogonal complement of M is equal to the null space of M. The author also considers using the family of subspaces of $\overset{\sim}{X}[\itex] that are intersections of one dimensional subspaces of X to prove the statement. SqueeSpleen Let [itex]X$ be an Inner Product Space. If for every closed subspace $M$, $M^{\perp \perp} = M$, then $X$ is a Hilbert Space (It's complete).
Hint: Use the following map: $T : X \longrightarrow \overset{\sim}{X}: T(y)=(x,y)=f(x)$ where $(x,y)$ is the inner product of $X$.

Relevant equations:
$S^{\perp}$ is always closed to every $S \subset X$

Attemp to solution.
I don't really know how to solve it, most theorems I have read, have Hilbert Space as a hyphotesis.
The only idea I had was trying to prove that $\overline{T(M^{\perp})} = M^{\circ}$ (Where $M^{\circ}$ is the Null Space of $M$).Edit: How do I do to have latex without jumping lines? I uploaded a pdf with a itext more closely formated to what I tried to write.

Edit2: I read other posts to find out how to do it, I only had to add an i to the "tex".

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I'm thinking about the family of subspaces of [itex] \overset{\sim}{X}[\itex] that are intersection of nullspaces of one dimensional subspaces of [itex]X[\itex], they are all closed. I think that if I can probe that they're all closed subspaces of [itex]X[\itex] I may, somehow, be able to prove the required statement.

What is a double orthogonal closed subspace inner product?

A double orthogonal closed subspace inner product is a type of inner product that is defined on two subspaces of a Hilbert space. It is a bilinear map that takes two vectors, one from each subspace, and returns a scalar value. This type of inner product is used to measure the angle between two subspaces and can be used to define a metric on the space of all subspaces.

How is this inner product related to Hilbert spaces?

Hilbert spaces are mathematical structures that are used to study vector spaces with an infinite number of dimensions. The double orthogonal closed subspace inner product is a tool that is used to define a metric on the space of all subspaces within a Hilbert space. It is also used in various applications, such as quantum mechanics and signal processing.

What are the properties of a double orthogonal closed subspace inner product?

Some of the key properties of this type of inner product include bilinearity, symmetry, and positivity. Bilinearity means that the inner product is linear in each of its arguments. Symmetry means that the inner product of two vectors is equal to the inner product of their conjugates. Positivity means that the inner product of a vector with itself is always greater than or equal to zero.

How is this inner product calculated?

The calculation of the double orthogonal closed subspace inner product involves taking the inner product of two vectors, one from each subspace, and then multiplying them together. This can be done using the standard inner product formula, which involves taking the complex conjugate of one vector and then multiplying the components together and summing them up.

What are some applications of this inner product?

The double orthogonal closed subspace inner product has a wide range of applications in mathematics and physics. It is used in quantum mechanics to define the concept of entanglement between two particles. It is also used in signal processing to measure the similarity between two signals. Additionally, it has applications in linear algebra, functional analysis, and optimization.

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