How to show $\|x-y\|_2^2 \leq \|x\|_2^2+2|x^Ty|$?

Question

Let $x,y \in \mathbb{R}^n$.

How can I show the following $$\|x-y\|_2^2 \leq \|x\|_2^2+2|x^Ty|$$

The above has been used by the authors of the following paper on page 8, in first line Online Principal Component Analysis.

Also, I think using the above for $M , N \in \mathbb{R}^{d \times k}$ the following is true. $$\|M-N\|_F^2 \leq \|M\|_F^2+2|\text{tr}(M^TN)|$$ where $\|\cdot\|_F$ is Frobenius norm.

Answer 1

You'll have hard time to prove:

$$\|x-y\|_2^2 \leq \|x\|_2^2+2|x^Ty|$$

Take $x = 0$ and $y \neq 0$, you get:

$$\|y\|_2^2 \le 0$$

which is unlikely.