alongside although knowing is enough to reconstruct because if then and otherwise ). In particular, the norm of should somehow be interpretable as the "norm of the hyperplane ". When then the Riesz representation theorem provides such an interpretation of in terms of the affine hyperplane as follows: using the notation from the theorem's statement, from it follows that and so implies and thus This can also be seen by applying the Hilbert projection theorem to and concluding that the global minimum point of the map defined by is The formulas provide the promised interpretation of the linear functional's norm entirely in terms of its associated affine hyperplane (because with this formula, knowing only the is enough to describe the norm of its associated linear ). Defining the infimum formula will also hold when When the supremum is taken in (as is typically assumed), then the supremum of the empty set is but if the supremum is taken the non-negative reals (which is the image/range of the norm when ) then this supremum is instead in which case the supremum formula will also hold when (although the atypical equality is usually unexpected and so risks causing confusion). Constructions of the representing vector Using the notation from the theorem above, several ways of constructing from are now described. If then ; in other words, This special case of is henceforth assumed to be known, which is why some of the constructions given below start by assuming Orthogonal complement of kernel If then for any If is a unit vector (meaning ) then (this is true even if because in this case ). If is a unit vector satisfying the above condition then the same is true of which is also a unit vector in However, so both these vectors result in the same Orthogonal projection onto kernel If is such that and if is the orthogonal projection of onto then Orthonormal basis Given an orthonormal basis of and a continuous linear functional the vector can be constructed uniquely by where all but at most countably many will be equal to and where the value of does not actually depend on choice of orthonormal basis (that is, using any other orthonormal basis for will result in the same vector). If is written as then and If the orthonormal basis is a sequence then this becomes and if is written as then Example