/sci/ - Science & Math

Pen and paper extends analogically a man's computational power. Through notes, sketches and registers, an individual can solve problems that he otherwise couldn't, such as the accounting of dozens of items, division of large degree polynomials, evaluation of convoluted integrals, solving differential equations, inversion of matrices, and even absolutely shitty things like multiplying big numbers.

Post neuralink zoomers most likely literally won't understand what this means, and this deeply saddens me. How do I cope?

>>11486057
Theorem 4: Any real or complex vector space admits a norm.
Proof: We choose a basis [math]B[/math] of V. For [math]v \in V[/math], we can write [math]v = \Sigma_{i=1} ^n k_i b_i[/math]. This is well defined, because the linear combination exists, is unique, and is finite. Set [math]f(v) = \Sigma_{i=1}^n |k_i|[/math]. [math]f(v)=0[/math] if and only if [math]v=0[/math] follows trivially. Similarly, [math]f( \lambda v) = \Sigma_{i=1}^n | \lambda k_i| = | \lambda | \Sigma_{i=1} ^n |k_i|[/math]. Finally, we have [math]u = \Sigma_{i=1}^n u_i b_i[/math] and [math]w = \Sigma_{i=1}^n w_i b_i[/math], where we’ve hidden the process of reindexing, etc. Then [math]f( u + w) = \Sigma_{i=1}^n |u_i+w_i| \leq \Sigma_{i=1}^n |u_i| + \Sigma_{i=1}^n |w_i| \leq f(u) + f(w)[/math].