Induced Norms And Basic Computations

From an Inner Product to a Length

An inner product compares two vectors and returns a scalar. If the same vector is placed in both slots, the resulting scalar measures its size in a way that depends on the chosen inner product. Taking the nonnegative square root gives the length induced by that inner product. The definition is short; accurate computations depend on knowing the inner product formula and, over \(\mathbb{C}\), using conjugation in the correct slot.

Let \(V\) be a vector space over \(\mathbb{F}\), where \(\mathbb{F}\) is \(\mathbb{R}\) or \(\mathbb{C}\). We use the convention that an inner product is linear in its first slot and conjugate-linear in its second slot. Thus, for \(a,b\in\mathbb{F}\) and \(\mathbf{u},\mathbf{v},\mathbf{w}\in V\), it satisfies \(\langle a\mathbf{u}+b\mathbf{v},\mathbf{w}\rangle=a\langle\mathbf{u},\mathbf{w}\rangle+b\langle\mathbf{v},\mathbf{w}\rangle\) and \(\langle\mathbf{u},a\mathbf{v}+b\mathbf{w}\rangle=\overline{a}\langle\mathbf{u},\mathbf{v}\rangle+\overline{b}\langle\mathbf{u},\mathbf{w}\rangle\). It also satisfies conjugate symmetry, \(\langle\mathbf{u},\mathbf{v}\rangle=\overline{\langle\mathbf{v},\mathbf{u}\rangle}\), and positive definiteness: \(\langle\mathbf{v},\mathbf{v}\rangle\) is a nonnegative real number, and it equals zero exactly when \(\mathbf{v}=\mathbf{0}\). Over \(\mathbb{R}\), conjugation has no effect, so the inner product is symmetric and linear in both slots.

The repeated vector is important: to find the length of \(\mathbf{v}\), calculate \(\langle\mathbf{v},\mathbf{v}\rangle\), not an inner product with some other vector. The inner product chosen for the space determines the calculation. In ordinary real coordinates it leads to a sum of squares; with complex coordinates it leads to squared magnitudes; for matrices and functions it leads to sums or integrals.

Two Basic Checks from the Definition

Before calculating examples, it is useful to establish two facts that follow directly from the inner product axioms. They justify the square root and show how the induced quantity responds to scaling. These facts do not require the triangle inequality or any other norm inequalities.

Proof. Positive definiteness says that \(\langle\mathbf{v},\mathbf{v}\rangle\) is a nonnegative real number. Its nonnegative square root therefore exists and is a nonnegative real number, so \(\|\mathbf{v}\|\geq 0\). If \(\mathbf{v}=\mathbf{0}\), then \(\langle\mathbf{0},\mathbf{0}\rangle=0\) by linearity, and hence \(\|\mathbf{v}\|=0\). Conversely, if \(\|\mathbf{v}\|=0\), then \(\langle\mathbf{v},\mathbf{v}\rangle=\|\mathbf{v}\|^2=0\). Positive definiteness then gives \(\mathbf{v}=\mathbf{0}\). \(\square\)

Proof. By linearity in the first slot and conjugate-linearity in the second, \(\langle c\mathbf{v},c\mathbf{v}\rangle=c\overline{c}\langle\mathbf{v},\mathbf{v}\rangle=|c|^2\|\mathbf{v}\|^2\). Both \(|c|\) and \(\|\mathbf{v}\|\) are nonnegative, so taking the nonnegative square root of both sides gives \(\|c\mathbf{v}\|=|c|\,\|\mathbf{v}\|\). This also covers \(c=0\), since both sides are then zero. \(\square\)

Coordinate Computations

For columns \(\mathbf{x},\mathbf{y}\in\mathbb{R}^n\), the standard dot product is \(\langle\mathbf{x},\mathbf{y}\rangle=\mathbf{x}^{\top}\mathbf{y}\). Thus the induced norm is the square root of the sum of the squares of the coordinates. In \(\mathbb{C}^n\), with the convention used here, the standard inner product is \(\langle\mathbf{x},\mathbf{y}\rangle=\sum_{j=1}^{n}x_j\overline{y_j}\). Setting \(\mathbf{y}=\mathbf{x}\) gives \(\sum_{j=1}^{n}x_j\overline{x_j}=\sum_{j=1}^{n}|x_j|^2\).

Matrices and Functions

The same definition applies when the vectors are matrices or functions, provided the space has an inner product. For real matrices, the Frobenius inner product of matrices of the same size is the sum of the products of corresponding entries. For complex matrices, the entry in the second matrix is conjugated. Consequently, placing a matrix in both slots gives the sum of the squared absolute values of all its entries. For functions with an integral inner product, placing a function in both slots gives the integral of its squared value in the real case, or its squared magnitude in the complex case.

A Reliable Computation Routine

Although the arithmetic differs across spaces, the sequence of decisions is the same. First identify the inner product that belongs to the space. Next substitute the same object in both slots. Then simplify the resulting scalar, taking care with conjugates or integration limits. Finally take the nonnegative square root, and check by squaring the result. The definition itself justifies this routine: the quantity to be square-rooted is exactly \(\langle\mathbf{v},\mathbf{v}\rangle\), and the positive-definiteness axiom guarantees that it is a nonnegative real number.

Common Mistakes

Forgetting complex conjugation can produce a value that is not even real. For the one-entry vector \(z=1+i\), a wrong calculation would use \(z^2=(1+i)^2=1+2i+i^2=2i\), then attempt to take \(\sqrt{2i}\) as though that were the squared length. The correction is to use the inner product with conjugation in the second slot: \(\langle z,z\rangle=z\overline{z}=(1+i)(1-i)=1-i+i-i^2=2\). Thus the induced norm is \(\sqrt{2}\), and \((\sqrt{2})^2=2\).

Do not stop at the squared length. In the function example, \(\frac{13}{3}\) is \(\langle f,f\rangle=\|f\|^2\), not the norm itself. The norm is \(\sqrt{\frac{13}{3}}\). Likewise, the matrix example gives squared length \(9\), while its induced norm is \(3\).

These computations also show why the particular inner product matters. A coordinate formula, a matrix formula, and an integral formula may look different, but each follows the same instruction: evaluate the chosen inner product on the object paired with itself, then take the nonnegative square root. Beyond these calculations, additional properties are needed to establish all the defining conditions of a norm.