[Verse 1]
Take two vectors from your space
Map them to a real number's place
Three sacred rules must always hold
Symmetry first, as we unfold
When u meets v, the same as v meets u
Linearity scales and adds on cue
[Chorus]
Inner product space, where angles dance
Dot product, integral - give structure a chance
Norm is the square root of self with self
Orthogonal means zero on the shelf
Cauchy-Schwarz keeps products in line
Triangle inequality's so divine
[Verse 2]
Standard form in R-n so clean
Sum of products, component scene
For functions on an interval's span
Integral of f times g, that's the plan
Positive definite, never below
Zero only when vector's zero
[Chorus]
Inner product space, where angles dance
Dot product, integral - give structure a chance
Norm is the square root of self with self
Orthogonal means zero on the shelf
Cauchy-Schwarz keeps products in line
Triangle inequality's so divine
[Bridge]
Project onto subspace W so bright
Sum of inner products scaled just right
Closest point theorem guarantees
Perpendicular error's what you see
Fundamental split of any vector
Projection plus orthogonal sector
[Verse 3]
Gram-Schmidt builds what nature lacks
Orthonormal basis from messy stacks
Take v-one, normalize to start
Subtract projections, work your art
Each new direction stands alone
Perpendicular paths you've grown
[Outro]
From dot products to function space
Inner structure shows its face
Mathematics' strongest inequality
Cauchy-Schwarz for all to see