An orthonormal basis of a Hilbert space is an orthonormal set whose linear span has the whole space as its closure. This completeness of an orthonormal system allows every vector to be expanded in the basis vectors, with convergence in norm. In finite dimension it is also a basis in the finite-linear-combination sense.
An orthonormal basis is a specific type of basis used in linear algebra and functional analysis that has two key properties: orthogonality and normalization. 1. **Orthogonality**: Vectors in the basis are orthogonal to each other. Two vectors \( \mathbf{u} \) and \( \mathbf{v} \) are said to be orthogonal if their dot product is zero, i.e.
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