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Let $X$ be a reflexive and separable Banach space. Let $(h_n)$ be a sequence dense in $\overline{B^*_1}$ (the closed ball in $X^*$ of radius $1$). Set $$ d(x,y) := \sum_{n=1}^\infty 2^{-n} |(x-y, h_n)|, $$ where $(x-y, h_n)=(x-y, h_n)_{X,X^*}$ is the dual pairing on $X$. Then $d$ is symmetric, satisfies the triangle inequality and $d(x,y) = 0 \implies x = y$, hence $d$ is a metric.

On closed bounded (by the norm) balls of $X$, the metric $d$ induces the weak topology.

Can the topology induced by $d$ be described in a functional way in relation with the weak topology? What is the relation with the bounded weak-topology or bounded weak$-\star$ topology? Would any answer change if $d$ is defined using the ratio $\frac{|(x-y,h_n)|}{1 + |(x-y,h_n)|}$ ?

Theorem 2.7.6 in Megginson's book says that the bounded weak${}^*$ topology is complete, that means there is a metric inducing the topology, is the metric of a similar form as above? the bounds weak${}^*$ topology on $X^*$ is different in nature to the bounded weak topology on $X$ as shown in Exercise 2.87-88.

This seems to be something that should be in Megginson's book on Banach spaces, I can only see something related in exercise 2.86-88 of section 2.7.

Let $X$ be a reflexive and separable Banach space. Let $(h_n)$ be a sequence dense in $\overline{B^*_1}$ (the closed ball in $X^*$ of radius $1$). Set $$ d(x,y) := \sum_{n=1}^\infty 2^{-n} |(x-y, h_n)|, $$ where $(x-y, h_n)=(x-y, h_n)_{X,X^*}$ is the dual pairing on $X$. Then $d$ is symmetric, satisfies the triangle inequality and $d(x,y) = 0 \implies x = y$, hence $d$ is a metric.

On closed bounded (by the norm) balls of $X$, the metric $d$ induces the weak topology.

Can the topology induced by $d$ be described in a functional way in relation to the weak topology or the bounded weak-topology? Would any answer change if $d$ is defined using the ratio $\frac{|(x-y,h_n)|}{1 + |(x-y,h_n)|}$ ?

This seems to be something that should be in Megginson's book on Banach spaces, I can only see something related in exercise 2.86-88 of section 2.7.

Let $X$ be a reflexive and separable Banach space. Let $(h_n)$ be a sequence dense in $\overline{B^*_1}$ (the closed ball in $X^*$ of radius $1$). Set $$d(x,y) := \sum_{n=1}^\infty 2^{-n} |(x-y, h_n)|\,$$ then $d$ is symmetric, satisfies the triangle inequality and $d(x,y) = 0 \implies x = y$, hence $d$ is a metric.

On closed bounded (by the norm) balls of $X$, the metric $d$ induces the weak topology.

Can the topology induced by $d$ be described in a functional way in relation with the weak topology? What is the relation with the bounded weak-topology or bounded weak$-\star$ topology? Would any answer change if $d$ is defined using the ratio $\frac{|(x-y,h_n)|}{1 + |(x-y,h_n)|}$ ?

Theorem 2.7.6 in Megginson's book says that the bounded weak${}^*$ topology is complete, that means there is a metric inducing the topology, is the metric of a similar form as above? the bounds weak${}^*$ topology on $X^*$ is different in nature to the bounded weak topology on $X$ as shown in Exercise 2.87-88.

This seems to be something that should be in Megginson's book on Banach spaces, I can only see something related in exercise 2.86-88 of section 2.7.

Let $X$ be a reflexive and separable Banach space. Let $(h_n)$ be a sequence dense in $\overline{B^*_1}$ (the closed ball in $X^*$ of radius $1$). Set $$ d(x,y) := \sum_{n=1}^\infty 2^{-n} |(x-y, h_n)|, $$ where $(x-y, h_n)=(x-y, h_n)_{X,X^*}$ is the dual pairing on $X$. Then $d$ is symmetric, satisfies the triangle inequality and $d(x,y) = 0 \implies x = y$, hence $d$ is a metric.

On closed bounded (by the norm) balls of $X$, the metric $d$ induces the weak topology.

Can the topology induced by $d$ be described in a functional way in relation with the weak topology? What is the relation with the bounded weak-topology or bounded weak$-\star$ topology? Would any answer change if $d$ is defined using the ratio $\frac{|(x-y,h_n)|}{1 + |(x-y,h_n)|}$ ?

Theorem 2.7.6 in Megginson's book says that the bounded weak${}^*$ topology is complete, that means there is a metric inducing the topology, is the metric of a similar form as above? the bounds weak${}^*$ topology on $X^*$ is different in nature to the bounded weak topology on $X$ as shown in Exercise 2.87-88.

This seems to be something that should be in Megginson's book on Banach spaces, I can only see something related in exercise 2.86-88 of section 2.7.