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False belief. If a family $(x_n)_{n\geq 1}$ is commutatively convergent (i.e. summable) in a normed space $(V,\|\ \|)$ then $$ \sum_{n\geq 1} \|x_n\|<+\infty\ . $$

This is true in finite dimensions and has counterexamples in infinite dimensions. Details and counterexamples can be found there.

Recall that a family $(x_i)_{i\in I}$ is called summable with sum $S$ iff $$ (\forall \epsilon>0)(\exists F\subset_{finite} I)(\forall F_1\subset_{finite} I)(F\subset F_1\Longrightarrow \|S-\sum_{i\in F_1}x_i\|<\epsilon) $$ This is equivalent with commutative convergence in case $I\subset \mathbb{N}$ is infinite.

False belief. If a family $(x_n)_{n\geq 1}$ is commutatively convergent (i.e. summable) in a normed space $(V,\|\ \|)$ then $$ \sum_{n\geq 1} \|x_n\|<+\infty\ . $$

This is true in finite dimensions and has counterexamples in infinite dimensions. Details and counterexamples can be found there.

Recall that a family $(x_i)_{i\in I}$ is called summable with sum $S$ iff $$ (\forall \epsilon>0)(\exists F\subset_{finite} I)(\forall F_1\subset_{finite} I)(F\subset F_1\Longrightarrow\\ \|S-\sum_{i\in F_1}x_i\|<\epsilon) $$ This is equivalent with commutative convergence in case $I\subset \mathbb{N}$ is infinite.

False belief. If a family $(x_n)_{n\geq 1}$ is commutatively convergent (i.e. summable) in a normed space $(V,\|\ \|)$ then $$ \sum_{n\geq 1} \|x_n\|<+\infty\ . $$

This is true in finite dimensions and has counterexamples in infinite dimensions. Details and counterexamples can be found there.

Recall that a family $(x_i)_{i\in I}$ is called summable with sum $S$ iff $$ (\forall \epsilon>0)(\exists F\subset_{finite} I)(\forall F_1\subset_{finite} I)(F\subset F_1\Longrightarrow ||S-\sum_{i\in F_1}x_i||<\epsilon) $$ This is equivalent with commutative convergence in case $I\subset \mathbb{N}$ is infinite.

False belief. If a family $(x_n)_{n\geq 1}$ is commutatively convergent (i.e. summable) in a normed space $(V,\|\ \|)$ then $$ \sum_{n\geq 1} \|x_n\|<+\infty\ . $$

This is true in finite dimensions and has counterexamples in infinite dimensions. Details and counterexamples can be found there.

Recall that a family $(x_i)_{i\in I}$ is called summable with sum $S$ iff $$ (\forall \epsilon>0)(\exists F\subset_{finite} I)(\forall F_1\subset_{finite} I)(F\subset F_1\Longrightarrow \|S-\sum_{i\in F_1}x_i\|<\epsilon) $$ This is equivalent with commutative convergence in case $I\subset \mathbb{N}$ is infinite.

False belief. If a family $(x_n)_{n\geq 1}$ is summable in a normed space $(V,\|\ \|)$ then $$ \sum_{n\geq 1} \|x_n\|<+\infty\ . $$

This is true in finite dimensions and has counterexamples in infinite dimensions. Details and counterexamples can be found there.

Recall that a family $(x_i)_{i\in I}$ is called summable with sum $S$ iff $$ (\forall \epsilon>0)(\exists F\subset_{finite} I)(\forall F_1\subset_{finite} I)(F\subset F_1\Longrightarrow ||S-\sum_{i\in F_1}x_i||<\epsilon) $$ This is equivalent with commutative convergence in case $I\subset \mathbb{N}$ is infinite.

False belief. If a family $(x_n)_{n\geq 1}$ is commutatively convergent (i.e. summable) in a normed space $(V,\|\ \|)$ then $$ \sum_{n\geq 1} \|x_n\|<+\infty\ . $$

This is true in finite dimensions and has counterexamples in infinite dimensions. Details and counterexamples can be found there.

Recall that a family $(x_i)_{i\in I}$ is called summable with sum $S$ iff $$ (\forall \epsilon>0)(\exists F\subset_{finite} I)(\forall F_1\subset_{finite} I)(F\subset F_1\Longrightarrow ||S-\sum_{i\in F_1}x_i||<\epsilon) $$ This is equivalent with commutative convergence in case $I\subset \mathbb{N}$ is infinite.