- Calculus, Volume 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability
- Tom M. Apostol
- Second Edition
- 1991
- 978-1-119-49676-2
1.13 Exercises
- Suppose we retain the first three axioms for a real inner product (symmetry, linearity, and homogeneity) but replace the fourth axiom by a new axiom $(4'):$ $(x, x) = 0$ if and only if $x = O.$ Prove that either $(x, x) \gt 0$ for all $x \neq O$ or else $(x, x) \lt 0$ for all $x \neq O.$
- Assume $(x, x) \gt 0$ for some $x \neq O$ and $(y, y) \lt 0$ for some $y \neq O.$ In the space spanned by $\{x, y\},$ find an element $z \neq O$ with $(z, z) = 0.$
- Incomplete.