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The article on Hilbert spaces has several examples of inner product spaces, wherein the metric induced by the inner product yields a complete metric space. An example of an inner product space which induces an incomplete metric is the space ([,]) of continuous complex valued functions and on the interval [,].
In the Hilbert space view, this is the orthogonal projection of onto the kernel of the expectation operator, which a continuous linear functional on the Hilbert space (in fact, the inner product with the constant random variable 1), and so this kernel is a closed subspace.
The first three functions in the sequence () = on [,].As converges weakly to =.. The Hilbert space [,] is the space of the square-integrable functions on the interval [,] equipped with the inner product defined by
Every inner product space is also a normed space. A normed space underlies an inner product space if and only if it satisfies the parallelogram law, or equivalently, if its unit ball is an ellipsoid. Angles between vectors are defined in inner product spaces. A Hilbert space is defined as a complete inner product space. (Some authors insist ...
The following, seemingly weaker, definition is also equivalent: Definition 3. A unitary operator is a bounded linear operator U : H → H on a Hilbert space H for which the following hold: the range of U is dense in H, and; U preserves the inner product of the Hilbert space, H. In other words, for all vectors x and y in H we have:
This example used the standard inner product, which is the map := ¯, but if a different inner product is used, such as := ¯ where is any Hermitian positive-definite matrix, or if a different orthonormal basis is used then the transformation matrices, and thus also the above formulas, will be different.
If a normal operator T on a finite-dimensional real [clarification needed] or complex Hilbert space (inner product space) H stabilizes a subspace V, then it also stabilizes its orthogonal complement V ⊥. (This statement is trivial in the case where T is self-adjoint.) Proof. Let P V be the orthogonal projection onto V.
The quotient space of by the vector subspace is an inner product space with the inner product defined by +, + := (),,, which is well-defined due to the Cauchy–Schwarz inequality. The Cauchy completion of A / I {\displaystyle A/I} in the norm induced by this inner product is a Hilbert space, which we denote by H {\displaystyle H} .