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quantum mechanics - Inner product of $\langle \phi | \psi \rangle . . . is generally complex-valued, as you saw, though it's easy to see that the inner product of any vector with itself is both real and non-negative In exactly the same way as R2, the magnitude of a vector | ψ ∈ C2 is ‖ψ‖ = √ ψ | ψ Just as before, the inner product of two unit vectors is typically not 1 (which would indicate that they were in fact the same vector)
quantum mechanics - Why does $\langle x | f \rangle = f (x . . . The i i -th component of vector v v → can be written as either vi v i, or v ⋅e^i v → ⋅ e ^ i, where e^i e ^ i is a unit vector pointing in the i i -th direction So you can say vi = e^i ⋅v v i = e ^ i ⋅ v → The equation f(x) = x|f f (x) = x | f is completely analogous to this, just that we are using an infinite dimensional (rigged) Hilbert space instead of a 'normal' vector space