orthogonality of basis of hilbert space

Question

Let $\{y_1,\ldots ,y_n\}$ be a sequence in a Hilbert space. Let $V_n$ be the linear span of $\{y_1,\ldots ,y_n\}$. If $||y_{n+1}||\le||{y-y_{n+1}}||\forall n$ and each $y\in V_n$, then can we say that $\langle y_i,y_j\rangle=0\ \forall i, j?$

I think yes, but unable to prove. Should we proceed in the direction of Gram-Schmidt orthogonilization? Thanks beforehand

Answer 1

Suppose the Hilbert space is defined over the real numbers. Let $j>i$. Then for all real $t$ $$ \|y_j \|^2\le \|t y_i-y_j\|^2 = |t|^2\| y_i\|^2 +2t \langle y_i,y_j\rangle+ \|y_j\|^2. $$ This yields in case $t\ne0$ to $$ -2 sign(t) \langle y_i,y_j\rangle \le |t| \cdot \| y_i\|^2. $$ Passing to the limit $t\searrow0$ and $t\nearrow0$ gives $\langle y_i,y_j\rangle =0$.

Similarly the proof works for complex Hilbert spaces.