- #1

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- Homework Statement
- Prove that ##\langle x, y \rangle = 0 \iff ||x =cy|| \geq ||x||## for all real c.

- Relevant Equations
- A few of them.

(We are working in a real Euclidean space) So, we have to show two things: (1)the arrow goes from left to right, (2) the arrow comes from right to left.

(1) if we're given ##\langle x, y \rangle = 0 ##

$$

|| x+ cy||^2 = \langle x,x \rangle + 2c\langle x,y\rangle +c^2 \langle y,y \rangle $$

$$

||x+cy||^2 = ||x||^2 + c^2||y||^2 +2c\langle x,y\rangle$$

$$||x+cy||^2 = ||x||^2 || + c^2||y||^2$$

As ## c^2 ||y||^2 \geq 0##, we have ## ||x+cy||^2 \geq ||x||^2##.

(2) If we're given that ## ||x+cy||^2 \geq ||x||^2##, all that we can do is expand ##||x+cy||^2## and proceed

$$

||x||^2 +c^2 ||y||^2 + 2c \langle x, y \rangle \geq ||x||^2 $$

$$

c^2 ||y||^2 + 2c \langle x, y \rangle \geq 0$$

But how to conclude that ## \langle x, y\rangle = 0##. This is one of the archetypical case when we have to move backwards in mathematics.

All I can do is to say, if ##c^2 ||y||^2 + 2c \langle x, y \rangle \geq 0## has to be true for all real c, the case may arise for some negative c such that ##2c \langle x, y \rangle ## may exceed ##c^2 ||y||^2 ## and so to rule out that possibility we must make ##\langle x, y \rangle = 0##. What do you say about that?

(1) if we're given ##\langle x, y \rangle = 0 ##

$$

|| x+ cy||^2 = \langle x,x \rangle + 2c\langle x,y\rangle +c^2 \langle y,y \rangle $$

$$

||x+cy||^2 = ||x||^2 + c^2||y||^2 +2c\langle x,y\rangle$$

$$||x+cy||^2 = ||x||^2 || + c^2||y||^2$$

As ## c^2 ||y||^2 \geq 0##, we have ## ||x+cy||^2 \geq ||x||^2##.

(2) If we're given that ## ||x+cy||^2 \geq ||x||^2##, all that we can do is expand ##||x+cy||^2## and proceed

$$

||x||^2 +c^2 ||y||^2 + 2c \langle x, y \rangle \geq ||x||^2 $$

$$

c^2 ||y||^2 + 2c \langle x, y \rangle \geq 0$$

But how to conclude that ## \langle x, y\rangle = 0##. This is one of the archetypical case when we have to move backwards in mathematics.

All I can do is to say, if ##c^2 ||y||^2 + 2c \langle x, y \rangle \geq 0## has to be true for all real c, the case may arise for some negative c such that ##2c \langle x, y \rangle ## may exceed ##c^2 ||y||^2 ## and so to rule out that possibility we must make ##\langle x, y \rangle = 0##. What do you say about that?