- 305

- 2

## Main Question or Discussion Point

In calculus on manifolds, within one of the problems of this book the dual space is indirectly defined. I'll quote:

Let [tex](R^n)^*[/tex] denote the dual space of the vector space [tex]R^n[/tex]. If [tex]x \in R^n[/tex], define [tex]\phi_x \in (R^n)^*[/tex] by [tex]\phi_x(y) = \langle x, y\rangle[/tex].

Now my problem is I just don't get it. Perhaps I have a preconcived idea of dual space that is messing me up (ie: hodge dual, or kernel of a subspace). The dual space description' in this problem doesn't appear to be anything like that.

Note that this is one of the problems in the first chapter of the book and the only sort of inner product discussed has been the [tex]R^n[/tex] euclian dot product.

Since only the euclian inner product has been mentioned if one wanted to talk about

[tex]\phi_x[/tex] above independent of what it is applied to one can rationalize this as a matrix operator for example:

[tex]

\phi_x(y) = \langle x, y \rangle = x^\text{T} y

[/tex]

so [tex]\phi_x[/tex] by itself in that sense is just.

[tex]

\phi_x = x^\text{T}

[/tex]

(Here [tex]\phi_x[/tex] is the matrix of the linear transformation).

If I was to describe the set of all such operators, I'd say that the dual space must then be the set of all the linear operators that map a vector to a number. Is that all this is?

Perhaps somebody can describe for me the application of the dual space as defined by this problem. Some context showing how this is used would help clarify the concept and it's significance.

Let [tex](R^n)^*[/tex] denote the dual space of the vector space [tex]R^n[/tex]. If [tex]x \in R^n[/tex], define [tex]\phi_x \in (R^n)^*[/tex] by [tex]\phi_x(y) = \langle x, y\rangle[/tex].

Now my problem is I just don't get it. Perhaps I have a preconcived idea of dual space that is messing me up (ie: hodge dual, or kernel of a subspace). The dual space description' in this problem doesn't appear to be anything like that.

Note that this is one of the problems in the first chapter of the book and the only sort of inner product discussed has been the [tex]R^n[/tex] euclian dot product.

Since only the euclian inner product has been mentioned if one wanted to talk about

[tex]\phi_x[/tex] above independent of what it is applied to one can rationalize this as a matrix operator for example:

[tex]

\phi_x(y) = \langle x, y \rangle = x^\text{T} y

[/tex]

so [tex]\phi_x[/tex] by itself in that sense is just.

[tex]

\phi_x = x^\text{T}

[/tex]

(Here [tex]\phi_x[/tex] is the matrix of the linear transformation).

If I was to describe the set of all such operators, I'd say that the dual space must then be the set of all the linear operators that map a vector to a number. Is that all this is?

Perhaps somebody can describe for me the application of the dual space as defined by this problem. Some context showing how this is used would help clarify the concept and it's significance.