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In the Royden, Fitzpatrick, Real Analysis, Theorem 9 in the section 15.4 is,

Theorem 9. Let $X$ be an infinite dimensional normed linear space. Then neither the weak topology on $X$ nor the weak-$*$ topology on $X^{*}$ is metrizable.

I am trying to understand the part that $X^{*}$ is not metrizable. To prove that the weak-$*$ topology on $X^{*}$ is not metrizable, he argue by contradiction. So assume that there is a metric $\rho^* : X^* \times X^* \to [0 ,\infty)$ that induces the weak-$*$ topology on $X^*$. And so he goes on to prove it, and in the last sentence he wrotes:

" .. Therefore $\mathcal{F} := \{\varphi_n \}$ is an unbounded sequence in $X^{*}$ that converges pointwise to $0$. This contradicts the Uniform Boundedness Theorem."

I don't understand this bold statement. Here I regard Uniform Boundedness Theorem as following statement ( p.269 of Royden's book ) :

The Uniform Boundedness Principle. For $X$ a Banach space and $Y$ a normed linear space, consider a family $\mathcal{F} \subseteq \mathcal{L}(X,Y)$. Suppose the family $\mathcal{F}$ is pointwise bounded in the sense that for each $x\in X$ there is a constant $M_x \ge 0$ for which $$ \|T(x)\| \le M_x \operatorname{for all} T\in \mathcal{F}.$$ Then the family $\mathcal{F}$ is uniformly bounded in the sense that there is a constant $M \ge 0$ for which $\|T \| \le M$ for all $T$ in $\mathcal{F}$.

To apply the Uniform Boundedness Principle, I think $X$ needs to become Banach space, but since $X$ may not be Banach space, I think that we cannot use directly the Uniform Boundedness Principle. Perhaps, metrizable infinite dimensional normed linear space is Banach ? Or should we attach the condition 'Banach' to the Theorem 9 above? How can we breakthrough this difficulty?

Plantation
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You have correctly pointed out the main issue in the proposed proof. In order to apply the uniform boundedness principle in the form you have stated, the normed space $X$ must be a Banach space. But as this hypothesis is not met, the proof does not work. This error is critical in the sense that the proof cannot be fixed and the statement that the dual of an infinite-dimensional normed space is not metrisable in the weak$^{*}$ topology is false. However, as you correctly stated, the result is true if the normed space $X$ is complete. See this post for more details.

Dean Miller
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    In fact, if $X$ is $c_{00}$, the space of eventually zero sequences, with the $\ell_2$ norm, then $X^= \ell_2$ isometrically; consider
    $ \ell_2 \ni {\lambda_n} \mapsto ({x_n} \mapsto \sum_{n=1}^\infty \lambda_n x_n) \in X^    $. Then the weak-star topology on $X^* = \ell_2 $ is the topology of pointwise convergence which is metrizable (it is generated by the seminorms $p_i( {\lambda_n}) = |\lambda_i|$.)     – Evangelopoulos Foivos Dec 04 '24 at 09:07
  • @EvangelopoulosFoivos : Can you explain why the topology generated by the seminorms $p_i $ is same as the topology of pointwise convergence more friendly? – Plantation Dec 04 '24 at 09:43
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    @Plantation By definition, a net $(x^a){a \in I}$ converges to $x$ w.r.t. to locally convex topology generated by the seminorms ${p_i}{i \ge 1}$ if and only if $p_i(x^a-x) \xrightarrow{a} 0$ for every $i\ge 1$. This basically means that $x^a$ converges to $x$ pointwise (i.e. in every coordinate). Since the family of seminorms is countable, this implies (by a known theorem) that the space is metrizable. – Evangelopoulos Foivos Dec 04 '24 at 10:43
  • @EvangelopoulosFoivos : I'm asking a question just to be sure. What's the notation $p_i(x^a -x)\xrightarrow{\text{a}}0 ( i \ge 1 ) $ exactly means? And why this implies that $x^{a}$ converges to $x$ pointwise? P.s. And for your last sentence in the comment we can refer to the Rudin's Functional analysis, 1.38 Remarks- (c) ( p.29 ) – Plantation Dec 04 '24 at 11:39
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    @Plantation By definition the seminorm $p_i(u):=|u_i|$ (the $i$-th coordinate) so that $p_i(x^a-x) := |x^a_i-x_i|$. So $p_i(x^a-x)\to 0$ for every $i$ if and only if $|x^a_i-x_i|\to0$ for every $i$ i.e. the net $x^a$ converges pointwise to $x$ – Evangelopoulos Foivos Dec 04 '24 at 12:44