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For linear operators $A_n$, $A$ in a finite dimensional vector space $V$, I am trying to prove the equivalence of

  1. $\|A_n - A\| = \sup_{x \in \mathbb{C}^n, |x| = 1} |A_nx-Ax| \to 0$ as $n \to \infty$ (Uniform Convergence; henceforth UC)
  2. $\forall x\in V: $ $|A_n x - Ax| \to 0$ as $n \to \infty$ (Strong Convergence; henceforth SC)
  3. $\forall x, y \in V: |\langle A_n x, y \rangle - \langle Ax, y \rangle | \to 0$ as $n \to \infty$ (Weak Convergence; henceforth WC)

I can show

  1. $(UC) \implies (SC)$: Let $x \in V$. Then $ x = \lambda y$ for $\lambda \in \mathbb{F}$ and $y \in V$ s.t. $|y| = 1$. Thus \begin{eqnarray*} |A_n x - Ax| = \lambda|A_n y - Ay|\leq \lambda\|A_n - A\| \to 0. \end{eqnarray*}
  2. $(SC) \implies (WC)$: by Cauchy-Schwarz, \begin{eqnarray*} |\langle A_n x, y \rangle - \langle Ax, y\rangle| = |\langle (A_n-A) x, y \rangle | \leq \langle (A_n-A)x, (A_n-A)x \rangle \langle y, y \rangle = |A_n x - Ax| \cdot |y| \to 0. \end{eqnarray*}

To establish equivalence, I also need $(WC) \implies (UC)$, but right now I have no clue how to show this. In particular, I assume I need to use finite-dimensionality somehow, but am not sure how.

PortMeadow
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    Hint: Without loss of generality assume $A_n$ is matrix and $A=0$. From (WC) follows that for any $\epsilon > 0$ each component of $A_n$ is eventually bounded by $\epsilon$. That is, $A_n$ converges pointwise to $0$. Now, two norms on the space of matrices are equivalent. Thus, we obtain (UC). – user251257 Aug 03 '15 at 15:02
  • That makes sense :) Do you have a reference for the proof that any two norms on the spaces of matrices are equivalent? (Hadn't covered that yet). – PortMeadow Aug 03 '15 at 16:05
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    two norms on a finite dimensional vector space are equivalent. should be in any analysis or calculus book. I am not sure whether it has name. – user251257 Aug 03 '15 at 16:07
  • usually, it is discussed after compactness and Heine Borel. – user251257 Aug 03 '15 at 16:45

1 Answers1

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Pulling down @user251257's comments.

Hint: Without loss of generality assume $A_n$ is a matrix and $A = 0$. From (WC) it follows that for any $\epsilon > 0$ each component of $A_n$ is eventually bounded by $\epsilon$. That is, $A_n$ converges pointwise to $0$. Now, any two norms on a finite dimensional vector space are equivalent. Thus, we obtain (UC).