Linear Functionals Review

# Linear Functionals Review

We will now review some of the recent material regarding linear functionals.

- On the
**Linear Functionals on Linear Spaces**page we said that if $X$ is a linear space then a**Linear Functional**on $X$ is a linear operator from $X$ to $\mathbb{C}$, and a**Real Linear Functional**on $X$ is a linear operator from $X$ to $\mathbb{R}$.

- On
**The Algebraic Dual of a Linear Space**page we defined the**Algebraic Dual**of a linear space $X$, denoted $X^{\#}$ to be the space of all linear functionals on $X$.

- We then proved an important result. We proved that if $X$ is a linear space and $\varphi \in X^{\#}$ and $x_0 \in X$ is such that $\varphi(x_0) \neq 0$ then:

\begin{align} \quad X = \ker \varphi \oplus \mathrm{span} (x_0) \end{align}

- On the
**Extensions of Linear Functionals on Subspaces of a Linear Space**page we proved that if $M \subset X$ is a subspace of $X$ and $\varphi \in M^{\#}$ then $\varphi$ can be extended to a linear functional $\psi \in X^{\#}$.

- On the
**Expressing a Linear Functional as a Linear Combination of Other Linear Functionals**page we proved that if $\varphi, \psi_1, \psi_2, ..., \psi_n \in X^{\#}$ then $\varphi$ is a linear combination of $\psi_1, \psi_2, ..., \psi_n$ if and only if:

\begin{align} \quad \bigcap_{k=1}^{n} \ker \psi_k \subseteq \ker \varphi \end{align}

- On
**The Topological Dual of a Normed Linear Space**page we defined the**Topological Dual**of a normed linear space $X$, denoted $X^{*}$ to be the space of all continuous linear functionals on $X$.

- We also noted that $X^{*}$ is always a Banach space.

- On the
**A Normed Linear Space is Finite-Dimensional IFF the Algebraic Dual and Topological Dual are the Same**page we proved a very important result which classifies finite-dimensional normed linear spaces. We proved that a normed linear space $X$ is finite-dimensional if and only if the algebraic dual of $X$ is the same as the topological dual of $X$, that is, $X^{\#} = X^*$. Hence, if $X$ is an infinite-dimensional normed linear space then there exists discontinuous linear functionals.