Proving $f - Pf \perp \text{span}\{e_j\}$ AND Bessel's Inequality Explained

Master two of the most fundamental proofs in functional analysis and advanced linear algebra! In this lecture, we give a rigorous, step-by-step proof showing that if Pf is the projection of a vector f onto a subspace spanned by an orthonormal system, then the residual vector (f - Pf) is strictly orthogonal to that entire span (f - Pf ⊥ span{ej}).

Building directly on this result, we also derive and explain Bessel's Inequality, showing you exactly how the norm of the projection relates to the norm of the original function.

📌 What You Will Learn in This Video:
0:00 - Introduction to the Projection Operator (Pf)
02:15 - Step-by-Step Proof: Proving f - Pf is Orthogonal to span{ej}
08:45 - What is Bessel's Inequality? (Concept & Formula)
11:30 - Full Derivation of Bessel's Inequality using the Projection Proof
18:15 - Geometric Interpretation & Why the Inequality Holds
22:00 - Summary & Key Exam Tips

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Видео Proving $f - Pf \perp \text{span}\{e_j\}$ AND Bessel's Inequality Explained канала Calculus Craze