Linear form

Indexing into a vector: The second element of a three-vector is given by the one-form ${\displaystyle [0,1,0].}$ That is, the second element of ${\displaystyle [x,y,z]}$ is $${\displaystyle [0,1,0]\cdot [x,y,z]=y.}$$

: The mean element of an ${\displaystyle n}$-vector is given by the one-form ${\displaystyle \left[1/n,1/n,\ldots ,1/n\right].}$ That is, $${\displaystyle \operatorname {mean} (v)=\left[1/n,1/n,\ldots ,1/n\right]\cdot v.}$$

Mean

: Sampling with a kernel can be considered a one-form, where the one-form is the kernel shifted to the appropriate location.

Sampling

of a net cash flow, ${\displaystyle R(t),}$ is given by the one-form ${\displaystyle w(t)=(1+i)^{-t}}$ where ${\displaystyle i}$ is the discount rate. That is, $${\displaystyle \mathrm {NPV} (R(t))=\langle w,R\rangle =\int _{t=0}^{\infty }{\frac {R(t)}{(1+i)^{t}}}\,dt.}$$

Net present value

If ${\displaystyle X}$ is a complex with a (complex) inner product ${\displaystyle \langle \,\cdot \,|\,\cdot \,\rangle }$ that is antilinear in its first coordinate (and linear in the second) then ${\displaystyle X_{\mathbb {R} }}$ becomes a real Hilbert space when endowed with the real part of ${\displaystyle \langle \,\cdot \,|\,\cdot \,\rangle .}$ Explicitly, this real inner product on ${\displaystyle X_{\mathbb {R} }}$ is defined by ${\displaystyle \langle x|y\rangle _{\mathbb {R} }:=\operatorname {Re} \langle x|y\rangle }$ for all ${\displaystyle x,y\in X}$ and it induces the same norm on ${\displaystyle X}$ as ${\displaystyle \langle \,\cdot \,|\,\cdot \,\rangle }$ because ${\displaystyle {\sqrt {\langle x|x\rangle _{\mathbb {R} }}}={\sqrt {\langle x|x\rangle }}}$ for all vectors ${\displaystyle x.}$ Applying the Riesz representation theorem to ${\displaystyle \varphi \in X^{\prime }}$ (resp. to ${\displaystyle \varphi _{\mathbb {R} }\in X_{\mathbb {R} }^{\prime }}$) guarantees the existence of a unique vector ${\displaystyle f_{\varphi }\in X}$ (resp. ${\displaystyle f_{\varphi _{\mathbb {R} }}\in X_{\mathbb {R} }}$) such that ${\displaystyle \varphi (x)=\left\langle f_{\varphi }|\,x\right\rangle }$ (resp. ${\displaystyle \varphi _{\mathbb {R} }(x)=\left\langle f_{\varphi _{\mathbb {R} }}|\,x\right\rangle _{\mathbb {R} }}$) for all vectors ${\displaystyle x.}$ The theorem also guarantees that ${\displaystyle \left\|f_{\varphi }\right\|=\|\varphi \|_{X^{\prime }}}$ and ${\displaystyle \left\|f_{\varphi _{\mathbb {R} }}\right\|=\left\|\varphi _{\mathbb {R} }\right\|_{X_{\mathbb {R} }^{\prime }}.}$ It is readily verified that ${\displaystyle f_{\varphi }=f_{\varphi _{\mathbb {R} }}.}$ Now ${\displaystyle \left\|f_{\varphi }\right\|=\left\|f_{\varphi _{\mathbb {R} }}\right\|}$ and the previous equalities imply that ${\displaystyle \|\varphi \|_{X^{\prime }}=\left\|\varphi _{\mathbb {R} }\right\|_{X_{\mathbb {R} }^{\prime }},}$ which is the same conclusion that was reached above.

Hilbert space

Discontinuous linear map

 – A vector space with a topology defined by convex open sets

Locally convex topological vector space

 – ordered vector space with a partial orderPages displaying wikidata descriptions as a fallback

Positive linear functional

 – Map from multiple vectors to an underlying field of scalars, linear in each argument

Multilinear form

 – Vector space with a notion of nearness

Topological vector space

(2015), Linear Algebra Done Right, Undergraduate Texts in Mathematics (3rd ed.), Springer, ISBN 978-3-319-11079-0

Axler, Sheldon

; Goldberg, Samuel (1980), "Chapter 4", Tensor Analysis on Manifolds, Dover Publications, ISBN 0-486-64039-6

Bishop, Richard

(1990). A course in functional analysis. Graduate Texts in Mathematics. Vol. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908.

Conway, John

Dunford, Nelson (1988). Linear operators (in Romanian). New York: Interscience Publishers.  0-471-60848-3. OCLC 18412261.

ISBN

(1974), Finite-Dimensional Vector Spaces, Undergraduate Texts in Mathematics (1958 2nd ed.), Springer, ISBN 0-387-90093-4

Halmos, Paul Richard

; Katznelson, Yonatan R. (2008), A (Terse) Introduction to Linear Algebra, American Mathematical Society, ISBN 978-0-8218-4419-9

Katznelson, Yitzhak

(1996), Linear algebra, Wiley-Interscience, ISBN 978-0-471-11111-5

Lax, Peter

; Thorne, Kip S.; Wheeler, John A. (1973), Gravitation, W. H. Freeman, ISBN 0-7167-0344-0

Misner, Charles W.

Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.  978-1584888666. OCLC 144216834.

ISBN

(1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.

Rudin, Walter

; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.

Schaefer, Helmut H.

(1985), "Chapter 3", A first course in general relativity, Cambridge, UK: Cambridge University Press, ISBN 0-521-27703-5

Schutz, Bernard

(2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.

Trèves, François

(2011), An Introduction to Manifolds, Universitext (2nd ed.), Springer, ISBN 978-0-8218-4419-9

Tu, Loring W.

(2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.

Wilansky, Albert