How to find dual spaces?

Question

I would greatly appreciate it if you could kindly share how to find dual spaces? For example, let X be the vector space of n-dimensional vectors with the Euclidean norm. Prove that X*=X. I know a dual space consists of all linear functionals on X. Wikipedia says that any linear functional in R^n can be written as a sum of the coordinates of a vector. How should I go from here?

Answer 1

Sketch: There's something called the dual basis of a vector space, given a basis. Let $\{u_1, u_2, ..., u_n\}$ be a basis of $X$; define a function $\epsilon_1$ by

$$\epsilon_1(a_1 u_1 + \dots + a_n u_n) = a_1$$

Check that this is a linear functional on $X$, and so $\epsilon_1 \in X^*$. Likewise define $\epsilon_2, \dots, \epsilon_n$, and then prove that $\{\epsilon_1, ..., \epsilon_n\}$ is a basis of $X^*$.

Answer 2

Take a linear functional $\varphi\in (\Bbb R^n)^{\ast}$. Let $\varphi_i=\varphi(e_i)$. Having written $x\in\Bbb R^n$ as $\displaystyle x=\sum_{i=1}^n x_ie_i$, observe $\varphi(x)=\displaystyle\sum_{i=1}^n x_i\varphi(e_i)=\sum_{i=1}^n x_i\varphi_i$ by linearity. It follows that if we identify $\varphi \leftrightarrow (\varphi_1,\ldots,\varphi_n)$ we can look at $\varphi(x)$ as a dot product $(x_1,\ldots,x_n)\cdot (\varphi_1,\ldots,\varphi_n)$ and consider $\varphi$ as a vector in $\Bbb R^n$.

We are tempted to consider the map $\varphi\in(\Bbb R^n)^\ast\mapsto (\varphi_1,\ldots,\varphi_n)\in\Bbb R^n$.

It is linear, for functionals are linear, so it is a linear transformation. Moreover, if $\varphi\mapsto (0,0,\ldots,0)$ it follows that $\varphi(e_i)=0$ for each $i$. Since $\varphi$ vanishes over a basis, it vanishes completely. Then our map is one-one and by finite dimension it is onto, hence an isomorphism.

You can also prove that $\dim (\Bbb R^n)^\ast=n$, whence having the same dimension as $\Bbb R^n$, it is isomorphic to it.