Let { v 1 , v 2 , v 3 , … } \{ {v_1},{v_2},{v_3}, \ldots \} be a sequence of elements of a Hilbert space, and suppose that (one or both of) the inequalities d 2 ∑ a i 2 ≤ ‖ ∑ a i v i ‖ 2 ≤ D 2 ∑ a i 2 {d^2}\sum {a_i^2 \leq {{\left \| {\sum {{a_i}{v_i}} } \right \|}^2} \leq {D^2}} \sum {a_i^2} hold for every finite sequence of scalars { a i } \{ {a_i}\} . If an element v 0 {v_0} is adjoined to { v i } \{ {v_i}\} , then the resulting set satisfies (one or both of) d 0 2 ∑ a i 2 ≤ ‖ ∑ a i v i ‖ 2 ≤ D 0 2 ∑ a i 2 d_0^2\sum {a_i^2 \leq {{\left \| {\sum {{a_i}{v_i}} } \right \|}^2} \leq D_0^2} \sum {a_i^2} , where, denoting the norm of v 0 {v_0} by r and its distance from the closed linear span of the v i {v_i} by δ \delta , \[ d 0 2 = d 2 + 1 2 ( r 2 − d 2 − ( r 2 + d 2 ) 2 − 4 d 2 δ 2 ) d_0^2 = {d^2} + \frac {1}{2}\left ( {{r^2} - {d^2} - \sqrt {{{({r^2} + {d^2})}^2} - 4{d^2}{\delta ^2}} } \right ) \] and \[ D 0 2 = D 2 + 1 2 ( r 2 − D 2 + ( r 2 + D 2 ) 2 − 4 D 2 δ 2 ) . D_0^2 = {D^2} + \frac {1}{2}\left ( {{r^2} - {D^2} + \sqrt {{{({r^2} + {D^2})}^2} - 4{D^2}{\delta ^2}} } \right ). \] Both bounds are best possible. If v 0 {v_0} is in the span of the original set, the expressions above simplify to d 0 = 0 {d_0} = 0 and D 0 2 = D 2 + r 2 D_0^2 = {D^2} + {r^2} . If the original set is a single unit vector v 1 {v_1} , so d = D = 1 d = D = 1 , and if v 0 ⊥ v 1 {v_0} \bot {v_1} is a unit vector so δ = 1 \delta = 1 , then the above is ( a 2 + b 2 ) ≤ ‖ a v 0 + b v 1 ‖ 2 ≤ ( a 2 + b 2 ) ({a^2} + {b^2}) \leq {\left \| {a{v_0} + b{v_1}} \right \|^2} \leq ({a^2} + {b^2}) , the Pythagorean Theorem. Several consequences are deduced. If v i {v_i} are unit vectors, ∑ a i 2 = 1 \sum {a_i^2 = 1} , and δ i {\delta _i} is the distance from v i {v_i} to the span of its predecessors (so that the volume of the parallelotope spanned by the v i {v_i} is V n = δ 1 δ 2 ⋯ δ n {V_n} = {\delta _1}{\delta _2} \cdots {\delta _n} ), the above result is used to show that ‖ ∑ i = 0 n a i v i ‖ ≥ V n / 2 n / 2 \left \| {\sum \nolimits _{i = 0}^n {{a_i}{v_i}} } \right \| \geq {V_n}/{2^{n/2}} .