## Franklin's Notes

### Dual space

If $X$ is a vector space over the field $\mathbb k$, then the dual space is the vector space $X^\ast=L(X;\mathbb k)$ of linear functionals on $X$, or linear transformations from $X$ to its underlying field. Addition of linear functionals in $X^\ast$ is given by and scalar multiplication is defined by

Additionally, linear transformations $A:X\to Y$ can be (naturally) transformed into linear transformations $A^\ast:Y^\ast\to X^\ast$ via precomposition. That is, given $f\in Y^\ast$, we define $A^\ast f=f\circ A$, which is linear because and The map $A^\ast$ is called the dual map of $A$.

# vector-space

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