Linear Algebra Done Right · Problem 6A.3

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · inner-products-and-norms

Problem

((a)) Show that the function taking an ordered pair

((x_1, x_2), (y_1, y_2))

of elements of

\mathbf{R}^2

|x_1 y_1| + |x_2 y_2|

is not an inner product on

\mathbf{R}^2

. ((b)) Show that the function taking an ordered pair

((x_1, x_2, x_3), (y_1, y_2, y_3))

of elements of

\mathbf{R}^3

x_1 y_1 + x_3 y_3

is not an inner product on

\mathbf{R}^3