Finite-Dimensional Linear Spaces Review

# Finite-Dimensional Linear Spaces Review

We will now review some of the recent material regarding finite-dimensional linear spaces.

- On the
**Equivalence of Norms in a Finite-Dimensional Linear Space**page we said that if $X$ is a normed linear space then two norms $\| \cdot \|_1$ and $\| \cdot \|_2$ on $X$ are said to be**Equivalent**if there exists $C, D > 0$ such that for all $x \in X$ we have that:

\begin{align} \quad C \| x \|_1 \leq \| x \|_2 \leq D \| x \|_1 \end{align}

- We then proved that if $X$ is a finite-dimensional linear space then ANY two norms defined on $X$ are equivalent Thus, the only "interesting" norms are those defined on infinite-dimensional linear spaces.

- On the
**Isometries on Normed Linear Spaces**page we said that if $X$ and $Y$ are normed linear spaces then a linear operator $T : X \to Y$ is an**Isometry**if for all $x \in X$ we have that:

\begin{align} \quad \| T(x) \| = \| x \| \end{align}

- We proved that every isometry is continuous and that every isometry is injective.

- On the
**Isomorphism Linear Operators on Normed Linear Spaces**we proved that if $X$ and $Y$ are normed linear spaces and $T : X \to Y$ is a bijective linear operator, then $T^{-1} : Y \to X$ is a linear operator.

- In particular, a bounded linear operator $T : X \to Y$ is said to be an
**Isomorphism**if $T^{-1} : Y \to X$ is also a bounded linear operator and the spaces $X$ and $Y$ are said to be isomorphic.

- We then proved that if $X$ and $Y$ are normed linear spaces then a bijective bounded linear operator $T : X \to Y$ is an isomorphism if and only if there exists $C, D > 0$ such that for all $x \in X$:

\begin{align} \quad C \| x \| \leq \| T(x) \| \leq D \| x \| \end{align}

- On the
**Two Finite-Dimensional Normed Linear Spaces of the Same Dimension are Isomorphic**page we proved that if $X$ and $Y$ are normed linear spaces and $\dim (X) = n = \dim (Y)$ then $X$ and $Y$ are isomorphic. We proved this by showing that any finite-dimensional linear space $X$ with $\dim (X) = n$ is isomorphic to $\mathbb{C}^n$.

- On the
**Every Finite-Dimensional Normed Linear Space is a Banach Space**page we proved that every finite-dimensional normed linear space is a Banach space.

- On the
**Riesz's Lemma**page we proved Riesz' lemma which states that if $X$ is a normed linear space and $Y \subset X$ is a proper closed subspace of $X$ then for all $\epsilon$ with $0 < \epsilon < 1$ there exists an $x_0 \in X$ with $\| x_0 \| = 1$ such that for all $y \in Y$:

\begin{align} \quad \| x_0 - y \| > 1 - \epsilon \end{align}