Finite-Dimensional Linear Spaces Review

Finite-Dimensional Linear Spaces Review

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

(1)
\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:
(2)
\begin{align} \quad \| T(x) \| = \| x \| \end{align}
  • We proved that every isometry is continuous and that every isometry is injective.
  • 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 $x \in X$:
(3)
\begin{align} \quad C \| x \| \leq \| T(x) \| \leq D \| x \| \end{align}
  • 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 $y \in Y$:
(4)
\begin{align} \quad \| x_0 - y \| > 1 - \epsilon \end{align}
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