# Conceptually, what is a Linear Functional?

Every site I see tells me, implementation-ally, what it is. For example, Wolfram MathWorld says: "A linear functional on a real vector space is a function T : V -> R, which satisfies the following properties..."

However, it just looks like a [linear] function...

I see that, in one case (I think...), a 'linear functional' could be defined as T(a,b) = 2a + b so I could say "a linear functional is a function which can be easily represented by a vector; then, applying this function to another vector is equivalent to an inner product defined as the dot product"

However, in a number of homework problems, they define something convoluted as seen in the solutions for 6.25 here: http://www2.engr.arizona.edu/~gehm/501/files/solutions/HW4%20Solutions.pdf

So, what IS a linear functional at is core? There HAS to be a simple explanation as to what it is that causes the details/notation to fall into place. However, everybody seems to work the other way around: starting with the muddy details and hoping the idea pops out of their butts.

There are two main, equivalent definitions.

Down-to-earth:

A linear functional is a map T : V -> K given by

T(c1, c2, ..., cn) = c1 * v1 + c2 * v2 + ... + cn * vn

where V is a vector space with field of scalars K, the v's are all fixed constants in K, and all the computations are with respect to some fixed basis. Obviously T(a, b) = 2a + b qualifies. Given any similar explicit function you can almost immediately say whether or not it's a linear functional. T(a, b) = ab isn't [though note you must prove that this cannot be written in the given form, even in a non-obvious manner], while T(a, b) = a certainly is. The constants themselves form a vector, <vi, v2, ..., vn>, in which case we can think of a linear functional as the function that dots an input vector with a fixed vector.

Abstract:

A linear functional on a vector space V with field of scalars K is a linear map T : V -> K. That is, it satisfies

T(v+w) = T(v) + T(w)

T(av) = aT(v)

[MathWorld specializes this to real and later complex vector spaces for no apparent reason. This version is better since it's used in the dual space construction.]

The homework solutions you linked uses a somewhat dense notation for determining the constants vi in the first definition I've written. Namely, if our fixed basis B is {e1, e2, ..., en}, we can find vi by computing T(ei) = T(0, ..., 0, 1, 0, ..., 0) where the 1 occurs in the ith place. That is,

T(c1, c2, ..., cn) = c1 * T(e1) + c2 * T(e2) + ... + cn * T(en)

Moreover, this can be written using a dot product as

T(c1, c2, ..., cn) = (c1, c2, ..., cn) dot (T(e1), T(e2), ..., T(en))

Or, getting rid of the coordinates,

T(c) = c dot (T(e1)e1 + T(e2)e2 + ... + T(en)en)

A linear functional on a vector space is mapping from the vector space to the field of scalars.

H:f---->H(f) belongs to F such that H is linear.

H(af+bg)=aH(f)+bH(g)

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RE:

Conceptually, what is a Linear Functional?

I can't find a good explanation of this anywhere.

Every site I see tells me, implementation-ally, what it is. For example, Wolfram MathWorld says: "A linear functional on a real vector space is a function T : V -> R, which satisfies the following properties..."

However, it...

a linear functional is a special KIND of linear function, one whose image(value) is a scalar.

let's look at your example functional:

T(a,b) = 2a + b.

we can write this another way:

T(a,b) = (2,1).(a,b), that is: T = (2.1).___

there is a natural basis for linear functionals on R^n:

{φ1, φ2,....,φn), where φj(x1,x2,....,xn) = xj.

in R^2, this "dual basis" consists of {φ1,φ2}, where

φ1(x,y) = x "pick the first coordinate"

φ2(x,y) = y "pick the second coordinate".

in this basis, we can write T as T = 2φ1+φ2, so T has "dual coordinates" (2,1)

(strange...that was the same vector in the dot product...hmm...a pattern maybe?).

linear functionals are common in mathematics.

one very well-known linear functional, on the vector space of integrable functions

f:[a,b]-->R, is the linear functional ∫[a,b] f, the definite integral of f from a to b.

conceptually, linear functionals are pretty simple: stick in a vector, get out a scalar.

because they are linear, they're "nice".

because linear functionals themselves form a vector space, we can use all the tools of linear

algebra on them. this space, is closely related to the domain space of the functionals themselves,

and called the dual space V* of V. the correspondence goes like this:

V............................V*

ej .............φj (also written as e^j, with a superscript)

coordinates........projections

since an inner product naturally gives one kind of linear functional <u,_>,

it should not be surprising that you can use linear functionals to in turn

define an inner product.

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