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Given two normed spaces, these have a topological structure. Then we can have the product topology on their product. I am assuming this is metrizable still. I know that we have $p$-norms on direct sums of normed spaces. Would the $p$-norms be the metric for this product topology? What happens if we have a collection of normed spaces, and we consider the product topology, what would the associative norm be?

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    Have a look at https://math.stackexchange.com/questions/464738/is-the-arbitrary-product-of-metric-spaces-a-metric-space, https://math.stackexchange.com/questions/168004/product-norm-on-infinite-product-space and https://math.stackexchange.com/questions/361778/show-that-the-countable-product-of-metric-spaces-is-metrizable. I think the quick answer is: combining metrics along the lines of the $p$-norm is fine for finite products: for countable products, you can get a metric that induces the product topology, but not a norm, for uncountable products you can't expect to get a metric. – Rob Arthan Jan 12 '24 at 21:44

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