# Definition A one-form $\tilde{\omega}$ is a linear, real-valued functional of [[Vector|vectors]], i.e. a one-form $\tilde{\omega}$ at $P \in M$, for some [[Manifold]] $M$, associates with a vector $\pmb{V}$ at $P$ a real number $\tilde{\omega}(\pmb{V}) \in \mathbb{R}$. One forms are linear: ^918ab6 $ \tilde{\omega}(a \pmb{V} + b \pmb{W}) = a \tilde{\omega}(\pmb{V}) + b \tilde{\omega}(\pmb{W}), \quad \forall a, b \in \mathbb{R}, \pmb{V}, \pmb{W} \in T_P $ And we define their addition and multiplication by real numbers as: $ (a \tilde{\omega})(\pmb{V}) = a[\tilde{\omega}(\pmb{V})] (\tilde{\omega} + \tilde{\sigma})(\pmb{V}) = \tilde{\omega}(\pmb{V}) + \tilde{\sigma}(\pmb{V}) $ One-forms live in the [[Cotangent Space]]. # Action on Vectors If we know the [[components]] of a one-form $\tilde{q}$, say $\{q_j\}$, and the components of a [[vector]] $\pmb{V}$, say $\{V^j\}$, then we can write the action of the one-form on the [[vector]] as: $ \tilde{q}(\pmb{V}) = \sum_j V^j q_j. $ # Examples of One-Forms The most common example of vectors is column vectors in matrix algebra, and then row vectors become one-forms since they map vectors into numbers. Another example is the [[Hilbert Space|Hilbert spaces]] used in Quantum Mechanics, where the [[Braket Notation|bra]] $\langle \phi|$ is the one-form and the [[Braket Notation|ket]] $\psi\rangle$ is the vector, and they map each other into numbers $\langle \phi | \psi \rangle$ # Tensor Algebra In the language of tensors, a one-form is a $(0,1)$ [[Tensor]].