如何在表单提交中使用jquery ajax

Changed the code to this now. Form is located in subscribe.php which in included into the index page.

<form class="subscribe" id="email" method="get">
    <input class="email" id="emailaddress" name="email" type="text" placeholder="EXAMPLE@EMAIL.COM" required>
    <input class="button" type="submit" id="emailSub" value="Subscribe">
</form>
<div id="subResult"></div>

Jquery is at bottom of the index page

$(document).ready(function() {
    $('#emailSub').click(function() {
        var email = $('#emailaddress').val();
        alert(email);
        $.ajax({
            type: "GET",
            dataType: 'script', //<-Data Type for AJAX Requests(xml | html | script | json)
            url: 'checksub.php',
            data: {
                'e': email
            },
            success: function(result) {
                $("#subResult").html(result);
            }
        });
    })
});

Checksub page is this now for testing.

<?php include 'admin/includes/db_connection.php'; ?>
<?php 
$email = mysqli_real_escape_string($db, $_GET['e']);
echo "test string:";
echo $email;
?>

maybe you should to change dataType? try it:

$.ajax({
    type: "GET",
    dataType: 'html', //<-Data Type for AJAX Requests(xml | html | script | json)
    url: 'checkemail.php',
    data: {
        'e': email
    },
    success: function(data) {
        $("#subResult").html(result);
    }
});

and

<input class="button" type="submit" id="emailSub" value="Subscribe">
                              ^ HERE

should be :

<input class="button" type="button" id="emailSub" value="Subscribe">

because of submit is submiting form and refreshing the page.. so jquery not work..

Specifying the Data Type for AJAX Requests

Are you preventing the default click event from firing submit?

$('#emailSub').click(function(e){
    e.preventDefault();
    var email = $('#emailaddress').val();
    $.ajax({url:"checkemail.php?e="+email,success:function(result){
        $("#subResult").html(result);
    }});
})