# cauchyschwartz

1. ### Cauchy-Schwartz inequality question

Let M ⊂ H be a closed linear subspace that is not reduced to {0}. Let f ∈ H,f /∈ M⊥. Prove that m = inf (f, u) is uniquely achieved. u∈M |u|=1 I have approached it with the cosine definition for inner product but that only works with Euclidean spaces. I want to know how to apply this...
2. ### [SOLVED] Cauchy-Schwartz and norms

Let ||-||_{1} and ||-||_{2} be the L^1-norm and the L^2-norm on C[a,b] the space of continuous real valued functions on the closed interval [a,b]: explicitly ||f||_{1}=\int_{a}^{b} |f|, ||f||_{2}=\sqrt{\int_{a}^{b} |f|^2}. Prove that ||f||_{1} \leq ||f||_{2} for all f \in C[a,b] (1). Hint...
3. ### Cauchy-Schwartz Inequality

Prove the Cauchy-Schwartz inequality |<u, v>| <= |u| |v|. (<= means less than or equal to) I attempted it a couple times, but didn't get anywhere. Help would be great, thanks.
4. ### Cauchy-Schwartz Inequality

This is a question that has been bugging me for quite some time. When I was a sophomore in Linear Algebra the Professor gave us a proof of the Cauchy Schwarz inequality like so, For any vectors u,v in R^n | u*v | = | ||u||*||v|| * cos (theta)| = ||u|| * ||v|| * |cos(theta)|...