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docs/mathematics/functional-analysis/normed-spaces/linear-functionals.md

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+# Linear functionals
+
+> *Definition 1*: a **linear functional** $f$ is a linear operator with its domain in a vector space $X$ and its range in a scalar field $F$ defined in $X$. 
+
+The norm can be a linear functional $\|\cdot\|: X \to F$ under the condition that the norm is linear. Otherwise, it would solely be a functional.
+
+> *Definition 2*: a **bounded linear functional** $f$ is a bounded linear operator with its domain in a vector space $X$ and its range in a scalar field $F$ defined in $X$. 
+
+## Dual space
+
+> *Definition 3*: the set of linear functionals on a vector space $X$ is defined as the **algebraic dual space** $X^*$ of $X$. 
+
+From this definition we have the following.
+
+> *Theorem 1*: the algebraic dual space $X^*$ of a vector space $X$ is a vector space.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+Furthermore, a secondary type of dual space may be defined as follows.
+
+> *Definition 4*: the set of bounded linear functionals on a normed space $X$ is defined as **dual space** $X'$.
+
+In this case, a rather interesting property of a dual space emerges.
+
+> *Theorem 2*: the dual space $X'$ of a normed space $(X,\|\cdot\|_X)$ is a Banach space with its norm $\|\cdot\|_{X'}$ given by
+>
+> $$
+>   \|f\|_{X'} = \sup_{x \in X\backslash \{0\}} \frac{|f(x)|}{\|x\|_X} = \sup_{\substack{x \in X \\ \|x\|_X = 1}} |f(x)|, 
+> $$
+>
+> for all $f \in X'$. 
+
+??? note "*Proof*:"
+
+    Will be added later.