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  1. Hilbert Space

    Linked via "parallelogram law"

    A Hilbert space ($\mathcal{H}$) is a complex vector space equipped with an inner product: $\langle \cdot, \cdot \rangle$ that allows for the definition of distance, and convergence. It is fundamental to functional analysis, quantum mechanics, and various branches of mathematical physics, serving as the mathematical arena where observable quan…

  2. Smooth Manifold

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    This tensor is fundamental to understanding the intrinsic geometry of the smooth manifold.
    The Torsion tensor $T(X, Y)$ measures the failure of the parallelogram law to close when defined by the connection:
    $$T(X, Y) = \nablaX Y - \nablaY X - [X, Y]$$
    A smooth manifold equipped with a connection that has zero torsion is called a symmetric manifold, a property often associated with Riemannian metrics that possess the Levi-Civita connection.

  3. Vectors (mathematics)

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    Vector Addition
    Vector addition is performed component-wise. Geometrically, this corresponds to the parallelogram law or the head-to-tail rule. If $\mathbf{u}$ and $\mathbf{v}$ are two vectors:
    $$\mathbf{u} + \mathbf{v} = \begin{pmatrix} u1 + v1 \\ u2 + v2 \\ \vdots \\ un + vn \end{pmatrix}$$