Bounded operators and adjoints

For a linear map T defined on all of a normed space H and taking values in another normed space K, bounded means there is one finite constant C with norm(Tx) at most C norm(x) for every x. The smallest such constant is the operator norm. Linearity turns continuity at zero into this uniform inequality, and the inequality implies continuity everywhere. Merely showing that each individual output has finite norm does not establish one constant that works for all inputs.

On square-summable sequences, the rule that multiplies coordinate n by 1/n is bounded with norm one. The rule multiplying coordinate n by n is not an everywhere-defined map from that space into itself: some square-summable inputs acquire nonsummable outputs. It can instead define an unbounded operator on a proper domain. Keeping that distinction prevents an invalid appeal to the closed graph theorem, which requires an everywhere-defined map between Banach spaces with closed graph.

For a bounded map T:H→K between Hilbert spaces, fix y in K. The expression inner-product(y,Tx) is a bounded linear functional of x under our convention. Riesz representation gives a unique vector T* y in H representing that functional. This defines an everywhere-defined bounded map T*:K→H. The defining identity is intrinsic, so the adjoint exists even without a coordinate basis or a finite matrix representation.

In orthonormal coordinates the adjoint matrix is the conjugate transpose. If coordinates carry a weighted inner product, blindly taking a transpose is wrong: the weight matrices enter the formula. Over real spaces no conjugation is needed, but the weights still matter. A bounded operator equals its adjoint when it is self-adjoint. A bounded operator is normal when it commutes with its adjoint. Neither property means that its matrix is diagonal in the coordinates currently displayed.

A vector is perpendicular to the range of T exactly when it belongs to the kernel of T*. Consequently the closure of the range of T is the orthogonal complement of that kernel. The closure cannot always be dropped: the diagonal operator with entries 1/n has dense but nonclosed range on square-summable sequences. Inverse problems can therefore have outputs arbitrarily close to attainable data even when the requested exact solution does not lie in the allowed input space.

For finite least squares, differentiating the squared residual gives the normal equation T*T x=T*b. In infinite dimensions a minimizer may fail to exist if the range is not closed. Even in finite dimensions, forming T*T squares a full-column-rank matrix's Euclidean condition number, so an identity useful for reasoning may be a poor numerical algorithm. Error amplification, solvability, and a stable computational method are related questions that still require separate checks.