Hilbert Space
- Normed Space are vector Spaces with a distance function
- Banach Space are special Normed Space which Cauchy Sequences converge
1 function space
- \(C[a,b]\) is the set of all functions from a to b is a linear operator.
- \(C^1[a,b]\) is the continuous function space
\[C^1[a,b] \subset C[a,b]\]
The continuous function space is also a subspace meaning it is closed under linear operations.
Vectors have a 0 vector so what is the equivalent for functions?
1.1 Zero function
Zero function maps all values to 0
2 Sequence Space
All Sequences that are Eventually 0 \(\subset\) All Sequences that Converge to 0 \(\subset\) All Sequences that Converge $
3 embed vector space into Hilbert space
To embed a vector space into a Hilbert space, you typically follow these steps:
Start with a Vector Space: Begin with a vector space, which is a set of vectors equipped with addition and scalar multiplication operations. The vector space can be finite-dimensional or infinite-dimensional.
Define an Inner Product: An inner product is a bilinear, symmetric, and positive-definite function that takes two vectors as input and returns a scalar. Introduce an appropriate inner product on the vector space.
Verify Completeness: Ensure that the vector space, equipped with the inner product, satisfies the completeness property. A Hilbert space is a complete inner product space, meaning that every Cauchy sequence (a sequence in which the elements get arbitrarily close to each other) in the space converges to a limit that also belongs to the space.
Embedding: The vector space can now be considered as embedded into the Hilbert space by treating the vectors in the original space as vectors in the Hilbert space. The inner product on the vector space is extended to the Hilbert space, allowing you to perform inner product operations on the embedded vectors.
Utilize Hilbert Space Properties: With the embedding, you can leverage the properties and tools available in the Hilbert space to analyze and operate on the vectors from the embedded vector space. This includes concepts like orthogonality, projections, and orthogonal complements.
It’s important to note that the process of embedding a vector space into a Hilbert space may vary depending on the specific context and requirements. The choice of inner product and the properties desired in the Hilbert space can influence the embedding process.
