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docs/mathematics/functional-analysis/inner-product-spaces/total-sets.md

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+# Total sets
+
+> *Definition 1*: a **total set** in a normed space $(X, \langle \cdot, \cdot \rangle)$ is a subset $M \subset X$ whose span is dense in $X$. 
+
+Accordingly, an orthonormal set in $X$ which is total in $X$ is called a total orthonormal set in $X$. 
+
+> *Proposition 1*: let $M \subset X$ be a subset of an inner product space $(X, \langle \cdot, \cdot \rangle)$, then
+>
+> 1. if $M$ is total in $X$, then $M^\perp = \{0\}$.
+> 2. if $X$ is complete and $M^\perp = \{0\}$ then $M$ is total in $X$.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+## Total orthornormal sets
+
+> *Theorem 1*: an orthonormal sequence $(e_n)_{n \in \mathbb{N}}$ in a Hilbert space $(X, \langle \cdot, \cdot \rangle)$ is total in $X$ if and only if 
+>
+> $$
+>   \sum_{n=1}^\infty |\langle x, e_n \rangle|^2 = \|x\|^2,
+> $$
+>
+> for all $x \in X$. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+> *Lemma 1*: in every non-empty Hilbert space there exists a total orthonormal set.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+> *Theorem 2*: all total orthonormal sets in a Hilbert space have the same cardinality.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+This cardinality is called the Hilbert dimension or the orthogonal dimension of the Hilbert space.
+
+> *Theorem 3*: let $X$ be a Hilbert space, then
+>
+> 1. if $X$ is separable, every orthonormal set in $X$ is countable.
+> 2. if $X$ contains a countable total orthonormal set, then $X$ is separable.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+> *Theorem 4*: two Hilbert spaces $X$ and $\tilde X$ over the same field are isomorphic if and only if they have the same Hilbert dimension.
+
+??? note "*Proof*:"
+
+    Will be added later.
+