Banach space

$\textbf{Banach space}$ A Banach space is a complete normed space, i.e. - a vector space $X$ over the real or complex numbers $\mathbb{F}$ with a norm $\lVert\cdot\rVert_X$ - which is complete, i.e. for every Cauchy sequence $(x_n)$ in $X$, there exists an element $x$ in $X$ such that $$\lim_{n\to\infty}\lVert x_n-x\rVert_X=0$$
$\textbf{Banach space}$ A Banach space is a complete normed space, i.e. - a vector space $X$ over the real or complex numbers $\mathbb{F}$ with a norm $\lVert\cdot\rVert_X$ - which is complete, i.e. for every Cauchy sequence $(x_n)$ in $X$, there exists an element $x$ in $X$ such that $$\lim_{n\to\infty}\lVert x_n-x\rVert_X=0$$
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𝐁𝐚𝐧𝐚𝐜𝐑 𝐬𝐩𝐚𝐜𝐞 A Banach space is a complete normed space, i.e. - a vector space 𝑋 over the real or complex numbers 𝔽 with a norm βˆ₯β‹…βˆ₯β‚“ - which is complete, i.e. for every Cauchy sequence (π‘₯β‚™) in 𝑋, there exists an element π‘₯ in 𝑋 such that limβ‚™β†’β€€Νš βˆ₯π‘₯β‚™βˆ’π‘₯βˆ₯β‚“=0
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