A Vector Space with a specified Inner Product. Theorem With V as a inner product space, ∀α,β∈V, ∀c∈F ∣∣cα∣∣=∣c∣∣∣α∣∣ ∣∣α∣∣>0,∀α=0 ∣⟨α∣β⟩∣≤∣∣α∣∣∣∣β∣∣ (Cauchy Schwarz Inequality) ∣∣α+β∣∣≤∣∣α∣∣+∣∣β∣∣